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Abstract
We study a system of coupled degenerate cavities with a switchable beam rotator embedded in the optical path of the main cavity. By exploiting the phase shift of the beam rotator dependent on the orbital angular momentum of the optical modes, and modulating the phase imbalance in the auxiliary cavity, it is shown that the system dynamics is equivalent to that of a charged particle in a 1D lattice subject to both static and time-dependent electrical fields. We investigate interesting physics and phenomena such as Bloch oscillations that arise due to the simulated electrical fields, and discuss how they can be used for practical purposes such as storing optical signals in a quantum memory. We also present a powerful measurement scheme to detect the system dynamics that is non-intrusive and technically easy to perform.
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1. Introduction
In recent years, orbital angular momentum (OAM) of light has generated intense interest in the research of quantum information and quantum simulation [1–9]. The optical OAM is associated with the winding number, $l$, for an optical mode with a phasor $e^{il\varphi }$ in a cylindrically symmetric optical system, where $\varphi$ is the azimuthal angle around the optical axis [10]. In theory, there is no upper limit on the OAM number, $l$, an interesting fact that gives OAM modes of light a unique advantage in studying quantum effects and physics in large-scale systems. It is then natural that it has been used for tasks such as demonstrating macroscopic quantum entanglement [1,2], improving the capacity of quantum and classical optical communication channels [5,11,12], and enhancing the security of holographic encryption [13].
By taking advantage of the OAM modes in degenerate optical cavities [14], we have shown in earlier work [7] that it is possible to perform sophisticated quantum simulation on nontrivial topological physics in a synthetic 2D system. Intriguingly, though the scale of the simulation is enhanced significantly due to the large number of available OAM modes in an optical cavity, the required physical resources and operational controls are drastically reduced because the degenerate OAM modes all reside in the same cavity. This idea has been extended in further studies to reduce the required physical resources and alleviate the technical challenges even more. For instance, it is shown in [8] how 1D topological effects can be demonstrated in a single degenerate cavity by ingeniously constructing a sharp boundary condition. In [9], the authors combine synthetic dimensions in both the OAM space and frequency domain to realize 2D quantum simulation in a single degenerate cavity. Also pursued are applications of optical OAM modes for other important quantum tasks and novel devices. In [4], we studied how to realize an all-optical quantum memory that stores the optical signal in the synthetic lattice by reducing the group velocity of a wave packet in the OAM space to 0 and thus freezing it. With the rapid experimental progress in constructing optical cavities supporting many OAM modes and realizing sophisticated manipulations on them [15–17], proliferate application of optical OAM in various quantum simulation and information tasks can be expected in the near future.
In earlier research of OAM-based quantum simulation [7], we have focused on simulating the effect of magnetic field in the synthetic lattice. From a quantum simulation point of view, it is interesting to explore the possibility of simulating electrical fields in the OAM space of light. Such capabilities can greatly extend the scope of photonic OAM research by enabling study of electrical field induced physics. Well known examples of such physics include Bloch oscillations and related effects, which we are interested in demonstrating in this work as a precursor to investigation of more exotic physics induced by electrical fields. Previously, Bloch oscillations caused by simulated electrical fields in optical systems have been studied in the frequency domain [18–21]. In [20], it is also shown that related effects like directional transport, dynamic localization, and super Bloch oscillation can be realized by using an array of electrodes to modulate a waveguide’s index of refraction both spatially and in time. Finding ways to simulate electrical fields in the OAM space will create a new playground for optical quantum simulation that allows to study Bloch oscillations and other electrical field enabled physics in a novel optical degree of freedom. In this work, we present optical systems and control protocols to achieve this goal. Due to the novel system design, we are able to fully exploit the inherent properties of optical OAM to realize very versatile physical configurations involving both static and time-dependent electrical fields, which allows to study Bloch oscillation and a variety of related effects like Bloch translation, dynamic localization, and super Bloch oscillation. Since all OAM modes reside in the same degenerate optical cavity, our system design is very simple and its physical size and complexity does not increase with the scale of the simulated system at all. Consequently, we not only have very easy operation that employs just a minimal number of controls for the simulation but can realize very precise and uniform electrical fields regardless of the size of the simulated system, as the field at each site in the simulated system is generated by the same optical element. Such advantages are very attractive for scalable quantum simulation, and can lead to reduced operational challenges than earlier schemes requiring arrays of control electrodes [20] which are subject to higher control complexity when the number of electrodes is increased to achieve higher precision in modulation and simulation.
Though our optical OAM system is attractive for scalable quantum simulation, it is nonetheless subject to several unique challenges that are less concerning in other physical systems. One notable such challenge is how to effectively and accurately measure the waveform in the OAM space, as measurement of OAM modes is one of the main difficulties in photonic OAM studies [1,5,22]. In order to overcome this challenge and detect the simulated Bloch oscillations, we introduce a novel method to determine the distribution of a wave packet in the OAM space without any OAM measurement, which not only serves our purposes very well but can be used as a general method for measuring any OAM distribution in a degenerate cavity. We further demonstrate the value of our scheme by showing that the simulated Bloch oscillation can be used for practical purposes such as functioning as a storage mechanism in an optical quantum memory.
2. Optical system
Our system is based on the coupled degenerate cavities shown in Fig. 1. Its design requirement and working principles have been described in detail in our earlier work [7]. In essence, the OAM modes in the main cavity, $\hat {b}_{l}$, are degenerate and coupled to adjacent modes $\hat {b}_{l\pm 1}$ due to coupling to the auxiliary cavity which has a pair of spatial light modulators in its optical path to increment or decrement the optical beam’s OAM number $l$. The coupling strength between adjacent OAM modes, $\kappa$, is determined by the free spectral range of the main cavity and the reflectivity of the coupling beam splitters [7]. By using a pair of electro-optical modulators (EOMs), it is also easy to tune $\phi$, the coupling phase between adjacent OAM modes [7].
Fig. 1. An optical system with a main degenerate cavity that supports many OAM modes, an auxiliary degenerate cavity that contains a pair of spatial light modulators (SLMs) and a pair of electro-optical modulators (EOMs) to couple adjacent OAM modes, and a switchable beam rotator (BR) in the optical path of the main cavity. The realization of the switchable BR is shown in Fig. 2.
The other crucial element in our optical system key to the simulation of electrical fields is a switching device between two optical paths in one of which a beam rotator (BR) is embedded. One way of realizing such a device capable of optional beam rotation in the main cavity’s optical path is to use a combination of Mach-Zehnder interferometers (MZIs) and Dove prisms. As shown in Fig. 2(a), two identical and cascaded 50:50 MZIs can be switched quickly using high-speed EOMs in one of their arms to make the interference between the two arms constructive or destructive. We use the same voltage for index modulation in the two MZIs such that their phases are locked. It can be easily seen that, in such an arrangement, though an incident optical signal can exit either of the two output ports of MZI$_{1}$ depending on its phase (0 or $\pi$ for constructive or destructive interference), it always exits the same output port of MZI$_{2}$ regardless of the phase of the MZIs. Therefore, by adjusting the phase of both MZIs to 0 or $\pi$, we can route the optical signal in two different paths internally while keep using the same input and output port external to the device. If we embed a BR in one of the two paths between MZI$_{1}$ and MZI$_{2}$, an optical beam in that path picks up additional phases. In this manner, the device in Fig. 2(a) functions as a switchable BR controlled by the phase of the MZIs.
Fig. 2. (a) A switchable beam rotation device constructed from two identical 50:50 Mach Zehnder inteferometers (MZIs) and a beam rotator (BR). The phases of the two MZIs are locked and an input optical signal will only travel in one of the two optical paths I and II. (b) The 50:50 MZI with an EOM to control the phase between its two arms. (c) The beam rotator constructed from a pair of Dove prisms with skewed optical axes relative to one another.
In Fig. 2(c), it is shown a realization for the BR using a pair of Dove prisms whose optical axes are rotated by $\theta /2$ with respect to each other. Since a Dove prism flips the transverse profile of any transmitted beam, the BR in Fig. 2(c) rotates a propagating optical beam by an angle $\theta$. The azimuthal phasor $e^{il\varphi }$ of the OAM mode $l$ then changes to $e^{il(\varphi +\theta )}$, which amounts to an additional phase shift of $l\theta$ proportional to the OAM number $l$. In combination with the MZIs in Fig. 2(a), the BR either has no effect, or gives an optical beam in the OAM mode $l$ a phase shift $l\theta$, depending on the phase of the MZIs.
When the phase of the MZIs in Fig. 2 is chosen such that the optical signal travels in path I and the BR is bypassed, it was shown in earlier work [7] that the optical system in Fig. 1 is conceptually equivalent to a particle in a translationally invariant 1D lattice. In this model, the OAM number $l$ is mapped to the spatial site number in the lattice. The coupling between the OAM modes corresponds to the hopping between neighboring sites in the lattice. The Hamiltonian of the system is thus [7]
where $\mathrm {h.c.}$ stands for Hermitian conjugate, and the tunneling phase $\phi (t)$ is determined by the imbalance between the two paths in the auxiliary cavity and can be tuned by the high-speed EOMs in Fig. 1. Assuming a constant coupling phase $\phi$ and introducing the Fourier transform $b_{q}=\sum _{l}e^{-iql}b_{l}$, $q$ the Bloch wave vector, we can easily obtain the energy spectrum of $H_{I}$,If we switch the phase of the MZIs in Fig. 2(a) to route the optical beam through path II, it will pick up an additional phase due to the BR. Because this phase is dependent on the OAM number, each site in the simulated lattice acquires an energy shift proportional to the site number $l$, just like a charged particle in a 1D lattice subject to a static electrical field. Since the number of times the beam passes the BR in unit time is proportional to the main cavity’s free spectral range $\Omega _{F}$, the energy shift in the simulated lattice is also proportional to $\Omega _{F}$. The system Hamiltonian reads
where $l\vartheta$ with $\vartheta =-\Omega _{F}\theta /2\pi$ is the effective electrostatic potential energy of the charged particle on site $l$. Therefore, with the help of the beam rotation device in Fig. 2, we can simulate a static electrical field $\vartheta$ in our optical system in Fig. 1 based on OAM of light. Also, notice that the coupling phase $\phi \left (t\right )$ can be related to the spatial integral of an effective vector potential. Under a gauge transformation that annihilates this effective vector potential, $H_{II}$ transforms to $H_{II}'=-\sum _{l}\kappa (b_{l}^{\dagger }b_{l+1}+h.c.)+\sum _{l}l(\vartheta +\frac { \partial \phi (t)}{\partial t})b_{l}^{\dagger }b_{l}$. It is seen that, by modulating the coupling phase $\phi (t)$, we can simulate an effective electrical field $-\partial \phi \left (t\right )/\partial t$, which can be time-dependent. Of much interest is periodic modulation of $\phi (t)$, which is appropriate for the EOMs to limit the voltage applied on them to ensure linear response. In particular, if we employ a sinusoidal modulation, we have $\phi (t) = \phi _0 + \phi _a \sin {\omega t}$, with $\phi _0$, $\phi _a$, and $\omega$ the DC offset, amplitude, and frequency of the modulation.It is worth noting that, by using the switchable BR in Fig. 2 and modulating the phase imbalance in the auxiliary cavity $\phi (t)$, we are able to simulate both static and time-dependent electrical fields in our system, using only a minimal number of optical elements and electro-optical modulations that do not increase with the size of the simulated system. Since they are realized by optical phases, there is little restriction on the magnitude and characteristics of the simulated fields. As will be shown in section 4, thanks to the versatile field configurations that can be realized in our system based on optical OAM, we can investigate a variety of interesting physical effects and phenomena depending on the characteristics of the simulated electrical fields.
3. System input
We now study the dynamics of an input wave packet as it is absorbed into the cavity and then moves in the OAM space under the simulated electrical fields. As discussed in earlier work [7], we can couple an incident optical signal into the cavity by using optical techniques such as graded coating to effectively introduce a low-reflectivity pinhole at the center of the input/output mirror. The size of the pinhole is chosen such that the non-helical $l=0$ mode in the cavity has significant coupling to outside modes but all other OAM modes have negligible leakage. This widely used technique [1,5,22] is based on the fact that all OAM modes are dark at the beam center except the $l=0$ mode which is the brightest at the center. We can describe the coupling of a wave packet into the cavity with the following Hamiltonian [23],
The optical signal outside of the cavity consists of the incoming input wave and a wave reflected by the input mirror and traveling in a new direction. We call it the transmitted wave. We first assume plane waves outside of the cavity, $A(x)=\Theta (-x)A(0-)e^{ikx}+\Theta (x)A(0+)e^{ikx}$, where $A(0-)$ and $A(0+)$ are the amplitudes for the incident and transmitted waves, $k$ is the wave vector relative to $\omega _{c}/c$, $\omega _{c}$ the main cavity frequency and $c$ the speed of light, and $\Theta (x)$ is the step function. Inside the optical cavity, we have $B_{l}=B_{0}e^{iq\left |l\right |}$, where $q$ is the Bloch vector determined by the energy relation
and where we have used the continuity condition $A(0)=\left [A(0+)+A(0-)\right ]/2$. We can then obtain the transmission amplitude $\varUpsilon$ and absorption amplitude $\varGamma$,Interestingly, if we choose the reflectivity of the pinhole such that the coupling coefficient $\xi ^{2}=4\kappa v_{a}\sin {q}$, we have $\varUpsilon =0$ and it is possible to achieve complete absorption into the cavity and hence a high write-in efficiency [4].
A wave packet input $\Theta (-x)\int f(k)e^{ikx}dk$ can be absorbed into the cavity [4] if its frequency profile $f(k)$ is limited to the cavity bandwidth $[-2\kappa,2\kappa ]$. The evolution of the cavity mode amplitude can be obtained by multiplying each scattering eigen function component with the time phasor for its energy. We then have
In the left most parts of Fig. 3, we plot the absorption process of a Gaussian input pulse into the cavity as obtained by numerically solving the time evolution under $H_{IO}$ in Eq. (4). Since the input wave packet has a Gaussian frequency profile centered around $q=\pi /2$, selecting a coupling coefficient $\xi =\sqrt {4\kappa v_{a}}$ results in optimal absorption. We can deduce from the dispersion relation in Eq. (9) that the absorbed pulse moves with a velocity $2\kappa$, as shown in Fig. 3.
Fig. 3. Numerically calculated time evolution of the power distribution in the cavity OAM modes normalized to the maximum intensity of the input pulse, when a Gaussian input pulse $e^{-(x-x_{0})^{2}/2\sigma ^{2}}$ with $x_{0}=-5\sigma$ and $\sigma = 2v_a/\kappa$, $v_a$ the speed of light in free space, is absorbed into the cavity under $H_{IO}$ in Eq. (4), and subsequently moves under $H_{II}$ in Eq. (3). The coupling coefficient of the pinhole is chosen to be $\xi =\sqrt {4\kappa v_{a}}$ to maximize the absorption. The phase imbalance in the auxiliary cavity is set to $\phi =0$ during the absorption process. (a) Bloch oscillation when the BR in Fig. 2 is switched in the optical path of the main cavity to simulate a static electrical field $\vartheta =0.8\kappa$. The white line is the theoretical result in Eq. (17). (b) Bloch translation when both a static field $\vartheta$ and a resonant modulation $\phi (t)=\phi _{a}\sin {\omega t}$ with $\phi _{a}=11$, $\omega =0.4\kappa$, and $\vartheta =2\omega$ are employed. The white line is the theoretical result for the mean OAM number in Eq. (19). (c) Dynamic localization when all parameters are the same as in (b) except the phase modulation amplitude $\phi _a = 11.625$ which is a zero for the Bessel function $J_2$. The white line is the value of the OAM when the fields are turned on. (d) Super Bloch oscillation when the periodic modulation $\phi (t)=-\frac {\pi }{2}+\phi _{a}\sin {\omega t}$ with $\phi _{a}=7.75$, $\omega =0.8\kappa$, and $\vartheta =1.7\kappa$, has a small beat frequency $\delta _{2}=2\omega -\vartheta =-0.1\kappa$. The white line is the theoretical result for the mean OAM number in Eq. (20).
4. Bloch oscillations and other physical effects induced by simulated fields
Once the input wave packet is absorbed into the cavity and reaches a predetermined mean position $\langle l\rangle$ in the OAM space, we can observe the effect of the simulated electrical fields by adjusting the phase of the MZIs in Fig. 2 to switch the BR into the optical path of the main cavity and/or modulating the phase imbalance $\phi (t)$ in the auxiliary cavity. The subsequent evolution of the wave packet in the OAM space is then governed by $H_{II}$ in Eq. (3) which gives rise to a time evolution operator [24]
with $\hat {L}=\sum _{l}lb_{l}^{\dagger }b_{l}$ the total OAM operator, $\hat {K}=\sum _{l}b_{l}^{\dagger }b_{l+1}$ the translation operator, andWe first consider the case $\vartheta (t)$ being a constant $\vartheta$ and $\phi (t)=0$, which is realized by switching the BR in the optical path of the main cavity and keeping the auxiliary cavity phase imbalance 0. In Fig. 3(a), we see that the OAM-number distribution of the wave packet oscillates with a period $T_{B}=2\pi /\vartheta$ which is the signature of Bloch oscillations [18,19,21,25] induced by a static electrical field. When Bloch oscillation occurs in the OAM lattice, the distribution of the wave packet is confined in the vicinity of a mean OAM number and oscillates around it. The amplitude of the oscillation is smaller with larger $\vartheta$. The fact that the wave packet returns to its original shape periodically can be easily deduced from Eq. (14) with $\eta (t)=\vartheta t$ and $\chi (t)=\frac {i\kappa }{\vartheta }(e^{-i\vartheta t}-1)$. After a time period of $t=mT_{B}$, $m$ an integer, we have $\eta (mT_{B})=2m\pi$ and $\chi (mT_{B})=0$, and hence $\hat {U}(mT_{B})=e^{-i2m\pi \hat {L}}=1$.
To observe the motion of a wave packet and appreciate the rich dynamics the system can exhibit under the simulated electrical fields, we investigate the time evolution of the mean OAM number, $\langle \hat {L}\rangle _{t}=\langle \sum _{l}lb_{l}^{\dagger }b_{l}\rangle _{t}$, where $\langle X\rangle _{t}$ denotes the average value of operator $X$ at time $t$. We have [24]
Next, we both employ a constant BR phase to generate a static $\vartheta$ and periodically modulate the phase of the auxiliary cavity such that $\phi (t)=\phi _{0}+\phi _{a}\sin {\omega t}$, $\omega$ the modulation frequency. In this case, effects of both static and periodic electrical fields are simulated. We have $\eta (t)=\vartheta t$ and
An interesting scenario arises in resonant driving $T=2\pi /\omega =nT_{B}=2\pi n/\vartheta$, or $\vartheta =n\omega$, $n$ an integer. In this case, $\delta _{n}=0$ and the $n$-th order term in Eq. (18) grows linearly with $t$ because $\lim _{x\to 0}\ sin{c} \ {x}=1$. It dominates over long time since all other terms are oscillatory in $t$. We then have $\chi (t)\approx \left (-1\right )^{n}\kappa J_{n}(\phi _{a}) e^{-i\phi _0}t$, which leads to a phenomenon known as Bloch translation [26] in which the wave packet’s OAM grows linearly with time,
In Fig. 3(b), the numerically calculated evolution is plotted along with the theoretical prediction in Eq. (19). The characteristics of the Bloch translation and the agreement between numerical and theoretical results are clearly visible.
An exception to the Bloch translation described above is when $\phi _a$, the amplitude of the modulation of $\phi (t)$, happens to be a zero of $J_n$ such that $J_n(\phi _a)=0$. When this is the case, we see from Eq. (19) that the mean OAM of the wave packet does not change with time. Physically, the wave packet is almost frozen in the OAM space as shown in the simulation in Fig. 3(c). It is a remarkable dynamic localization effect in which a wave packet initially confined to a limited number of sites will not spread to an infinite distance over time [27].
If the driving is not exactly resonant, but with a small detuning $\delta _{n}=n\omega -\vartheta \ll \omega$, the system dynamics are quite different. Since the value of ${\textrm{sinc}} {x}$ is the largest for small $x$ and drops quickly when $x$ increases, the $n$-th order term again dominates in Eq. (18) and we have $\chi (t)\approx 2\left (-1\right )^{n}\frac {\kappa }{\delta _{n}}J_{n}(\phi _{a})e^{ i\left (\delta _{n}t/2-\phi _{0}\right )}\sin {\frac {\delta _{n}t}{2}}$. According to Eq. (16), it leads to
Interestingly, we find that the wave packet oscillates in the OAM space with the beat frequency $\delta _{n}$ as shown in Fig. 3(d). The amplitude of this oscillation, $\kappa /\delta _{n}$, is much larger than that of the ordinary Bloch oscillation without the periodic driving, $\kappa /\vartheta$, since $\delta _{n}\ll \vartheta$. For this reason, it has been called super Bloch oscillation.
5. Application in optical quantum memory
Aside from its fundamental interest, the Bloch oscillations studied in this work can be put into practical use in novel quantum devices. In [4], it is shown how to realize an all-optical quantum memory in the degenerate cavity system by controlling the movement of an optical pulse. In this application, once the optical signal has been absorbed into the cavity, it is crucial to have a reliable scheme to store it that prevents it from moving in the OAM space without bound and allows to retrieve it in its original shape later. In [4], we reduce the group velocity of the wave packet in the OAM space to 0 and thus effectively freeze it by annihilating the coupling strength $\kappa$. This is realized using optical interference and it requires additional optical elements [4].
Interestingly, Bloch oscillation provides an alternative storage mechanism. As shown in Figs. 4(a) and (b), when the optical signal has been absorbed into the cavity and moved to a predetermined mean OAM number at a group velocity of $2\kappa$, we turn on a simulated static electrical field $\vartheta$ as described in section 4. Though the wave packet is not frozen and keeps evolving, the mean OAM value does not increase linearly with time any more, even though the coupling $\kappa$ remains on. Instead, the motion of the wave packet in the OAM space is bound and periodic due to Bloch oscillations, and the optical memory enters the storage phase. The amplitude of the oscillation can be suppressed by increasing $\vartheta$. To read the stored optical signal out, we turn off $\vartheta$ and switch the coupling phase to $\phi =\pi$ when the wave packet has completed an integer number $m$ of Bloch oscillations in the cavity in a time period of $mT_B$. According to the dispersion relation in Eq. (2), doing so will cause the wave packet to move toward the $l=0$ mode at a group velocity of $2\kappa$, as shown in Fig. 4(b). Once the wave packet reaches the $l=0$ mode, it will leak out of the cavity via the same pinhole that was used for system input [4]. In fact, the read-out process is the time reversal of the input process discussed in section 3 and is governed by the same physics [4]. As shown in Fig. 4(a), we will get an optical pulse out of the cavity in the same profile with the input signal [4].
Fig. 4. Bloch oscillations as a storage mechanism for an optical quantum memory based on the degenerate cavity system. (a) Control sequence for the phase imbalance $\phi (t)$ in the auxiliary cavity and the simulated electrical field $\vartheta$, as well as the calculated output power at $x=0$ normalized to the maximum intensity of the input pulse specified in Fig. 3(b). Evolution of the power distribution in the OAM space under the control sequence in (a). (c) The same simulation as in (a) except that photon loss is taken into account. The same photon loss model as in [4] is used in the calculation with a decay rate of $\gamma _l = 0.2\kappa e^{-\lvert l \rvert } + 0.01\kappa$ for OAM mode $l$. (d) The same simulation as in (b) except that the photon loss rates in (c) are taken into account.
Suitable physical parameters for the OAM-based optical quantum memory design have been given in [4]. Assuming cavity mirrors with focal lengths on the order of centimeters which results in an FSR of $2\pi \times 0.5$GHz-$2\pi \times 1.0$GHz, we can obtain a system bandwidth $4\kappa$ of $2\pi \times 50$MHz-$2\pi \times 100$MHz. In reality, the optical signal will decay due to photon losses caused by the optical elements in the system. Using the same photon loss model as in [4], we simulate the write-in, storage, and read-out process and plot the results in Figs. 4(c) and (d).
Obviously, we can also take advantage of the dynamic localization and super Bloch oscillation in Figs. 3(c) and (d) for storage of optical signals in the cavity. The analysis and simulation is similar to that for Bloch oscillation and is omitted.
6. Detection
In order to observe the spectacular phenomena described in section 4 that occur in the degenerate cavity and investigate their underlying physics, effective and reliable detection techniques capable of revealing essential characteristics of the physical effects without disturbing the system dynamics are needed. A straightforward option is to make one of the cavity mirrors slightly lossy and allow a small portion of the light to leak out of the cavity via it. This leaked light is a snapshot of the wave packet in the cavity with the same OAM distribution but much weaker intensity. The OAM distribution of the leaked light can be determined by established photonic OAM measurement techniques. For instance, one can use an $l$-th order SLM to shift the OAM number of all modes in the leaked light by $l$ [1,5,22], after which the $l$-th OAM mode is converted to the 0-th mode whose intensity is concentrated at the beam center. By using a single-mode fiber or pinhole [1,5,22], one can then separate it from other modes and measure its intensity. By repeating this measurement for different OAM number $l$ and at different times, it is possible in principle to measure the time evolution of the OAM distribution depicted in Fig. 3. Unfortunately, introducing a lossy mirror will have a significant adverse impact and is thus a severe drawback.
We introduce a much less intrusive yet very efficient technique to detect the Bloch oscillation and other interesting phenomena in the cavity. Instead of inserting a lossy mirror in the cavity system, we read the wave packet out of the cavity for measurement. It is accomplished much like described in section 5, except that, in order to trace the time evolution of the Bloch oscillation, we would need to turn off the simulated electrical field $\vartheta$ at an arbitrary moment during a Bloch oscillation cycle and make the wave packet move toward the $l=0$ mode for exit, instead of doing so only when the wave packet has completed an integer number $m$ of Bloch oscillations. Unfortunately, the need for measurement at an arbitrary time $t= mT_B + \tau$ ($m$ an integer) creates a subtle problem because the wave vector $q_\pm$ of the positive and negative OAM branch (see Fig. 4(b)) of the wave packet may have deviated from $\pi /2$ and $3\pi /2$ since the simulated electrical field $\vartheta$ causes the wave vector to change linearly with time such that
When the wave packet undergoes the Bloch oscillation between $mT_B$ and $mT_B + \tau$, the positive and negative branch of the OAM distribution accumulates a phase
In section 5, we read the stored wave packet out of the cavity when it has completed an integer number $m$ of Bloch oscillations. At such a moment, $\delta _\pm (mT_B) = 0$, and the relative phase between the positive and negative OAM branch is the same value when the optical pulse was absorbed into the cavity. Because the relative phase has been preserved, we can get an output pulse in the same profile with the input when the two branches travel back to $l=0$ and recombine, as shown in Figs. 4(a) and (b). However, for a Bloch oscillation time $\tau$ not an integer multiple of $T_B$, $\delta _\pm$ is generally not 0, which means that the relative phase between the two branches has changed by $2\delta (\tau )$ from the value when the pulse was absorbed. Naively making them travel back to $l=0$ and recombine to be read out will generally not produce an output signal in the same profile of the input as in Fig. 4(a), because the interference condition between the two branches has changed. In the extreme situation, it may not even be possible to read the signal out of the cavity if the interference between the two branches are close to being completely destructive.
In order to overcome the difficulty associated with the phases $\delta _\pm (\tau )$, we change the value of the phase imbalance in the auxiliary cavity to $\phi = q_+(\tau ) + \pi = \frac {3\pi }{2} - \vartheta \tau$ when the simulated field $\vartheta$ is turned off as shown in Fig. 5(a). According to the dispersion relation in Eq. (2), the energy of the positive and negative OAM branch becomes $E(q_\pm ) = \pm 2\kappa$, and the group velocity becomes $v_\pm = 0$. Therefore, both the positive and negative OAM branches are frozen in the OAM space (except for slight dispersion due to their finite bandwidth) but their phases keep changing at a constant rate determined by $E(q_\pm ) = \pm 2\kappa$. If we keep $\phi$ at $q_+(\tau ) +\pi$ for a time period of
we find that the phase accumulated during $\varDelta \tau$ cancels $\delta _\pm (\tau )$ in Eq. (22) and the positive and negative branches are in phase again. At this moment, we set the phase imbalance in the auxiliary cavity to $\phi = \pi - \vartheta \tau$, as indicated in Fig. 5(a). From the dispersion relation in Eq. (2), we see that the wave packet will start moving toward $l=0$ at a group velocity of $2\kappa$, as shown in Fig. 5(b). Since the wave packet did not move during the time interval $\varDelta \tau$, the difference of its OAM at the moment $\tau + \varDelta \tau$ from the DC value of the Bloch oscillation, $\overline {\langle l \rangle }$, is just the displacement of the Bloch oscillation at time $\tau$. According to Eq. (17), we haveAs shown in Fig. 5(b), after we set $\phi = \pi - \vartheta \tau$ to make the wave packet move toward $l=0$, the time it takes for it to reach $l=0$ and be read out of the cavity is (relative to the average time $\overline {\langle l \rangle }/2\kappa$)
According to Eq. (24), this time delay is a direct representation of the displacement of the Bloch oscillation at time $\tau$. We can then detect the Bloch oscillation in the cavity by measuring the time delay $\varDelta t$, which is accomplished straightforwardly by monitoring the light intensity at the pinhole output position $x=0$ to see when the output pulse emerge. Notice that such a measurement does not involve any OAM manipulation and thus can be performed very easily and reliably, a major advantage of our detection scheme. It is also a general protocol to measure any OAM distribution in the degenerate cavity by mapping the OAM displacement to a time delay, not limited to detection of Bloch oscillations.
Fig. 5. Scheme to read out the wave packet at a time $mT_B + \tau$ during a Bloch oscillation cycle. The parameters are $\vartheta = 0.4\kappa$, $m=1$, and $\tau = 0.4T_B$. (a) Control sequence for the phase imbalance $\phi (t)$ in the auxiliary cavity and the simulated electrical field $\vartheta$, as well as the calculated output power at $x=0$. (b) Evolution of the power distribution in the OAM space under the control sequence in (a). (c) The same as in (a) except that photon loss rates in Fig. 4(c) are taken into account. (d) The same as in (b) except that photon loss rates in Fig. 4(c) are taken into account.
In Fig. 6(a), we plot the calculated output light intensity at $x=0$ as a function of time since the wave packet starts moving toward $l=0$ for different values of $\tau$. It is clearly seen that the time it takes for the signal to emerge at the output, which can be characterized by the time for the output power to reach the maximum, is an oscillatory function of $\tau$ with a period of $T_B$. This oscillatory behavior is a direct representation of the Bloch oscillation in the cavity according to our analysis. We can go one step further to reconstruct the OAM distribution in the cavity at time $\tau$ from the measured output waveform by using the input-output relationship studied in section 3 to retrace the read-out process. Therefore, our detection scheme is a very powerful yet non-intrusive and technically easy method to measure every detail of the system dynamics without any loss of information. As shown in Fig. 6(b), photon loss does not prevent us from detecting the Bloch oscillation as long as the power of the signal remains strong enough for it to be measured.
Fig. 6. (a) Calculated output power at $x=0$ using the control sequence in Fig. 5(a) for different values of $\tau$. The horizontal axis is the time delay since the wave packet in the cavity starts moving toward $l=0$ at a group velocity of $2\kappa$. (b) The same simulation except that photon loss rates in Fig. 4(c) are taken into account.
7. Conclusion
In summary, we have shown how we can simulate electrical fields in an optical system based on degenerate OAM modes of light by using BRs and modulating the coupling phase. We study the rich dynamics that arise due to the versatile driving conditions that can be realized in our system. We demonstrate that the resulting Bloch oscillations can be used as a storage mechanism in an optical quantum memory, and introduce a very effective yet non-intrusive measurement scheme to detect the system dynamics.
Funding
National Natural Science Foundation of China (11974334); National Key Research and Development Program of China (2017YFA0304103, 2017YFA0304504).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef]
2. M. Krenn, M. Malik, M. Erhard, and A. Zeilinger, “Orbital angular momentum of photons and the entanglement of Laguerre-Gaussian modes,” Philos. Trans. R. Soc., A 375(2087), 20150442 (2017). [CrossRef]
3. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014). [CrossRef]
4. X.-W. Luo, X. Zhou, J.-S. Xu, C.-F. Li, G.-C. Guo, C. Zhang, and Z.-W. Zhou, “Synthetic-lattice enabled all-optical devices based on orbital angular momentum of light,” Nat. Commun. 8(1), 16097 (2017). [CrossRef]
5. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008). [CrossRef]
6. F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8(1), 15516 (2017). [CrossRef]
7. X.-W. Luo, X. Zhou, C.-F. Li, J.-S. Xu, G.-C. Guo, and Z.-W. Zhou, “Quantum simulation of 2D topological physics in a 1D array of optical cavities,” Nat. Commun. 6(1), 7704 (2015). [CrossRef]
8. X.-F. Zhou, X.-W. Luo, S. Wang, G.-C. Guo, X. Zhou, H. Pu, and Z.-W. Zhou, “Dynamically Manipulating Topological Physics and Edge Modes in a Single Degenerate Optical Cavity,” Phys. Rev. Lett. 118(8), 083603 (2017). [CrossRef]
9. L. Yuan, Q. Lin, A. Zhang, M. Xiao, X. Chen, and S. Fan, “Photonic Gauge Potential in One Cavity with Synthetic Frequency and Orbital Angular Momentum Dimensions,” Phys. Rev. Lett. 122(8), 083903 (2019). [CrossRef]
10. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
11. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
12. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]
13. X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14(2), 102–108 (2020). [CrossRef]
14. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8(1), 189–196 (1969). [CrossRef]
15. Z-D. Cheng, Z-D. Liu, X-W. Luo, Z-W. Zhou, J. Wang, Q. Li, Y-T. Wang, J-S. Tang, J-S. Xu, C-F. Li, and G-C. Guo, “Degenerate cavity supporting more than 31 Laguerre-Gaussian modes,” Opt. Lett. 42(10), 2042–2045 (2017). [CrossRef]
16. Z-D. Cheng, Q. Li, Z-H. Liu, F-F. Yan, S. Yu, J-S. Tang, Z-W. Zhou, J-S. Xu, C-F. Li, and G-C. Guo, “Experimental implementation of a degenerate optical resonator supporting more than 46 Laguerre-Gaussian modes,” Appl. Phys. Lett. 112(20), 201104 (2018). [CrossRef]
17. Z-D. Cheng, Z-H. Liu, Q. Li, Z-W. Zhou, J-S. Xu, C-F. Li, and G-C. Guo, “Flexible degenerate cavity with ellipsoidal mirrors,” Opt. Lett. 44(21), 5254–5257 (2019). [CrossRef]
18. C. Bersch, G. Onishchukov, and U. Peschel, “Experimental observation of spectral Bloch oscillations,” Opt. Lett. 34(15), 2372–2374 (2009). [CrossRef]
19. L. Yuan and S. Fan, “Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring oscillator,” Optica 3(9), 1014–1018 (2016). [CrossRef]
20. C. Qin, L. Yuan, B. Wang, S. Fan, and P. Lu, “Effective electric-field force for a photon in a synthetic frequency lattice created in a waveguide modulator,” Phys. Rev. A 97(6), 063838 (2018). [CrossRef]
21. H. Chen, N. Yang, C. Qin, W. Li, B. Wang, T. Han, C. Zhang, W. Liu, K. Wang, H. Long, X. Zhang, and P. Lu, “Real-time observation of frequency Bloch oscillations with fibre loop modulation,” Light: Sci. Appl. 10(1), 48 (2021). [CrossRef]
22. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef]
23. D. Roy, C. M. Wilson, and O. Firstenberg, “Colloquium: Strongly interacting photons in one-dimensional continuum,” Rev. Mod. Phys. 89(2), 021001 (2017). [CrossRef]
24. H. J. Korsche and S. Mossmann, “An algebraic solution of driven single band tight binding dynamics,” Phys. Lett. A 317(1-2), 54–63 (2003). [CrossRef]
25. L. Du and Y. Zhang, “Light splitting and stopping and their combination via controllable Bloch oscillation in a lattice,” J. Opt. Soc. Am. B 37(7), 2045–2052 (2020). [CrossRef]
26. W-H. Hu, L. Jin, and Z. Song, “Dynamics of one-dimensional tight-binding models with arbitrary time-dependent external homogeneous fields,” Quantum Inf. Process. 12(11), 3569–3585 (2013). [CrossRef]
27. D. H. Dunlap and V. M. Kenkre, “Dynamic localization of a charged particle moving under the influence of an electrical field,” Phys. Rev. B 34(6), 3625–3633 (1986). [CrossRef]
