# Optical and spin-optical superpositions modulated by Aharonov-Bohm effect

###### Abstract

Generation of Aharonov-Bohm (AB) phase has achieved a state-of-the-art in mesoscopic systems with manipulation and control of the AB effect. The
possibility of transfer information enconded in such systems to non-classical states of light increases the possible scenarios where
the information can be manipulated and transfered. In this paper we propose a quantum transfer of the
AB phase generated in a spintronic device, a topological spin transistor (TST), to an quantum optical device, a coherent state superposition
in high-Q cavity and discuss optical and spin-optical superpositions in the presence of an AB phase. We demonstrate that the AB phase generated in the TST can be transfered to the coherent state superposition, considering the interaction with the spin state and the quantum optical manipulation of the coherent state superposition. We show that these cases provide examples of two-qubit states modulated by AB effect and that the phase parameter can be used to control the degree of rotation of the qubit state. We also show under a measurement on the spin basis, an optical one-qubit state that can be modulated by the AB effect. In these cases, we consider a dispesive interaction between a coherent state and a spin state with an acquired AB phase and also discuss a dissipative case where a given Lindblad equation is achieved and solved.

Keywords: Quantum transfer; Aharonov-Bohm phase; Spintronics; Quantum Optical Devices.

###### Contents

- 1 Introduction
- 2 TST and coupling to high-Q cavity device
- 3 Quantum Transfer of the Aharonov-Bohm phase to the coherent state superposition
- 4 Bit-enconding of information with AB phases
- 5 Storage of bits enconded by AB phases
- 6 Qudits from enconded states by quantum transfered AB phases
- 7 One-qubit state modulated by AB phase
- 8 Two-qubit spin-optical state modulated by AB phase
- 9 Spin-optical density operator modulated by AB phase, dissipation and Lindblad equation
- 10 Conclusions
- 11 Acknowledgements

## 1 Introduction

A typical ilustration of the Aharonov-Bohm (AB) effect [1] is the imprinting of a phase factor on the electron’s wave function in the region outside a solenoid under a magnetic field applied only in its interior. Another important feature of this effect is that this phase factor, the so-called AB phase, generated in the presence of a non-zero gauge potential in a region with a null electromagnectic field modifies the corresponding energy spectrum of the charged particle [2]. The presence of the AB phase represents a direct influence of a vector potential on the particle dynamics and one of remarkable possibilities of quantum interference [3].

The emergence of the AB effect in the problem of a charged particle minimally coupled to a gauge field , () , where , for a singly connected region, with , has as a consequence the splitting of the wave function in two parties with the acquired AB phase term [4, 5]. Considering a magnetostatic field, the whole path closed and the gauge fixing contribution of the AB phase shift term is given by . The interference in the electron states coming from the right and left parts around the solenoid give rise to a superposition state with the acquired AB phase given by This can be undertaken from an interaction between a source-field and the electron, as discussed in [6], with the whole state given by , colapsing to a final configuration where the left and right states are identical except for the presence of the AB phase, and , leaving the total state in the following final form .

Methods of generating AB phases have growth in different fields, representing an important role in solid-state interferometers [7, 8], nanotubes [9], transmission microscopy [10], AlSb/InAs heterostructures [11], quantum Hall effect regime [12], Kondo resonance [13] and spin transport [14]. Detection of this effect in photonic states [15], optical induction in mesoscopic systems [16] and interaction of AB ring in a high-Q cavity [17, 18, 19] are examples of the increasing importance of such effect in interacting light-and-matter systems and optical analogues of the AB effect [20].

Advances in the domain of artificial gauge fields have allowed new possibilities in the application of AB effect and geometrical phases [21]. Optical Berry phase [22], Aharonov-Casher effect [23, 24] as well as the AB effect for neutral particles have also been explored in several contexts [25, 27, 28, 26]. The possibility of controlling the AB effect in different scenarios and retransmit the phase from the domain where the effect was originally created is an important step in the phenomenological control and the manipulation of the associated information. Such a realization is particularly important for quantum information and computation tasks. Gate controlled AB oscillations [38] and topological spin transistor (TST) [30] are advances in such direction. The last one makes use of quantum spin Hall insulators, experimentally realized in HgTe quantum wells [32], with particular importance in spintronics applications [36]. Analog of a Datta-Das spin transistor [35], in the TST the properties of the quantum spin Hall insulator (QSHI) are used to generate an AB phase, that makes and an effective rotation of the outgoing spin state with respect to the ingoing spin [30]. The control of AB phases [33] can also be used for bit codification of the information associated to the AB effect. In the case of a TST, the presence or absence of flux is associated to the presence or not of a phase in the outgoing electron state.

Here, we discuss optical and spin-optical superpositions in carrying an AB phase. We start considering the instance of a TST as spintronic device to implement a quantum transfer to a coherent state in a high-Q cavity as quantum optical device. Furthermore, we show that manipulation and control of the AB effect in a TST allows the generation of spin superpositions with an AB phase while the interaction with a coherent state in a high-Q cavity leads to the possibility of manipulating this phase in the quantum optical scenario. We consider a setup consisting of a TST coupled to a high-Q cavity carrying initially a non-classical state of light, a coherent state, and a detector that project the final spin state in one specific state (see figure LABEL:cavity). Coupling a quantum ring to high-Q cavities have been considered previously in other contexts [17]. The high-Q cavity is built experimentally using supermirrors with extremelly low losses, achiving Q factor of the order . The integration with a TST can be realized in the implementation with for instance a photonic cristal cavity [31].The AB phase generated in the TST can be quantum transfered to the coherent state superposition, considering the interaction with the spin state and the quantum optical manipulation of the coherent state superposition. We demonstrate that the AB phase generated in the TST can be transfered to the coherent state superposition, considering the interaction with the spin state and the quantum optical manipulation of the coherent state superposition. We will show that these cases provide examples of two-qubit states modulated by AB effect and the phase parameter can be used to control the degree of rotation of the qubit state. We also discuss the fact that a flux control can then be associated to a binary codification, allowing a sequence (string) of bits of a given length be generated in such a way. In particular, the AB phase can be quantum transfered from the spintronic device to a quantum optical one, increasing the possible scenarios where the information can be manipulated and stored. Finally, We will discuss a particular dissipative scenario, by means which we provide and solve a given Lindblad equation for coupling the coherent state and a spin state with an acquired AB phase to a thermal revervoir.

This work is organized as follows: In Sec. II, we discuss the generation and bit-enconding of AB phases by exploring the AB effect. In Sec III, we propose the quantum transfer of AB phase to a coherent state superposition. In Sec. IV, we discuss the use of quantum transfer to store a string of bits enconded by AB phases. Sec. V is reserved to our concluding remarks.

## 2 TST and coupling to high-Q cavity device

When the outgoing spin state is injected dispersivelly in the high-Q cavity, the interaction between the coherent state and the spin state, in a dispersive regime, can generate an entanglement. The setup starts with the electron in the ferromagnet lead being injected into QSHI ring in a -spin polarized direction, . Due the properties of the QSHI, the state splits in the ring setup according to the helical properties of the edge states, the edge states of opposite spin polarization counterpropagate in the of spin up ( clockwise direction) and spin down ( counterclockwise direction) , in the top and bottom edges of the ring, respectivelly [30]. The presence of a magnetic flux into the middle of the ring inserts a AB phase , with , in the outgoing electron. This electron state is rotated in the plane leading to a state of the form

(1) |

When the outgoing state is injected in a high-Q cavity where there is a defined coherent state , with a dispersive interaction of the outgoing state with the coherent state in an off-resonant regime, the coupling interaction in the high-Q cavity can be written by

(2) |

where is an effective coupling and the number operator is associated to the quantized electromagnectic field, with and the creation and annihilation operators acting on the coherent state and the single-electron operator given by

(3) |

The quantized electromagnetic field in a high-Q cavity initially in a coherent state interacts effectivelly with the electron states and , where a degeneracy is assumed, corresponding to electron states of same kinectic energy. The single-mode field initially in the coherent state has an associated frequency , such that a energy difference between the electron kinectic energy and the single-mode frequency is given . A prototipical ilustration of such scheme is given in the figure 1, where a spin state is injected in the -direction and a superpostion is generated along ring in a superposition of edge states in up and down direction. The AB phase is acquired by the outgoing state that interacts with a high-Q cavity state.

For large detuning and short evolving times, where the magnetostatic flux is much slower than the frequency of the quantized electromagnetic field, the dispersive interaction between the AB-shifted electrons and the single-mode field can be considered. In order to see that the interaction (2) in fact leads to a phase entanglement with the coherent state for an appropriate time of interaction, we can act with the Hamiltonian

(4) | |||||

(5) | |||||

(6) | |||||

(7) | |||||

(8) | |||||

(9) |

In the general case, it follows

(10) |

The unitary evolutions are then given explicitly by

the terms implies the presence of a sign contribution.

(11) | |||||

(12) |

As a consequence, we can write the unitary evolutions as

(13) |

(14) |

## 3 Quantum Transfer of the Aharonov-Bohm phase to the coherent state superposition

In the case of the electron superposition carrying the AB phase the state after interaction is the following

(15) | |||||

In order to project the whole state in a non-classical state of light carrying the AB phase, a detector measures the electron state in one of the spin polarizations or .

We can rewrite eq. (15) with the terms of AB phase directly in the coherent state superposition,

(16) | |||||

This state can also be simplified in the form

(17) | |||||

where is related to by means of . If after interaction the spin polarization is detected in the state , the non-classical state is projected in the following superposition of coherent states carrying the AB phase

(18) |

On the other hand, if the measurement is realized in the spin polarization , we will have

(19) |

We can specify the time of interaction in , adjusting the high-Q cavity parameters, such that we now have for each case

(20) |

The coefficients in (20) can be rearranged by the action of a Hadamard-type gate operation in the basis of coherent states [34],

(21) |

considering the projection relations for coherent states and . After this gate operation the state takes the form

(22) | |||||

Under the condition of negligible overlap, the coherent states and can be considered orthonormal, , being described by a Hilbert space of the two-level system spanned by and . The resulting states are

(23) | |||||

Simplifying for each field state, apart a normalization factor, we have

(24) |

that corresponds to a selective measurement of the right-handed electron state and

(25) |

to left-handed electron state . In both cases, the AB phase is transfered to the coherent state superposition.

## 4 Bit-enconding of information with AB phases

One possible use of the generation and control of AB phases is the binary codification. We can propose a bit-enconding in the generation of AB phases by means of the flux control in the solenoid, in particular, we can control the imprinting or not of the phase, by absence or not of the flux, due to the switch-on or switch-off of the flux in the solenoid. Since the absence of the gauge potential will lead to absence of the AB phase, the off-state can be characterized by a null vector potential, while the on-state can be characterized by its presence. This implies in a bit defined from absence of AB phase, an off-state, and the presence of AB phase, the on-state: . This can be characterized formally by the binary function dependent of presence of a AB phase or its absence:

(26) |

Consequently, a sequential generation of AB phases stands for a sequence of the following type , where corresponds to the -th order of -th measured phase term. In a string of bits, this results in

(27) |

In particular, this codification can be used to implement any ASCII code or a given quantity of bytes. For instance, the string of bits corresponds to the sequence

(28) |

This binary codification using AB phases could consequently be implemented in any computer device where the presence of the AB effect could be controlled in a integrated way with other components. This could allow any binary message and also contrasts with other possibilities of codification based on chirality or flux orientation.

## 5 Storage of bits enconded by AB phases

Bits codified by superpositions of states carrying AB phases are given by the following state

(29) |

These states can be stored in appropriate quantum devices, as long as storage times and system interaction be slower than the life time of state. Appropriate unitary evolutions, as in the operator

(30) |

will leave the codified bit invariant

(31) |

as long as the evolution time is a period . As an example, for the state (28),

(32) |

If we have parallelized high-Q cavities or multimode cavities the transfer can also be implemented to store the state in the form

(33) |

corresponding to a string of bits stored in parallelized high-Q cavities in coherent state superpositions.

The cavities can be used to store sequentially the bit information with the generation of a product state. The information enconded in the string of bits can be retransfered to other system or detected by measurement of the interference pattern associated to the state. It is important to note that this state can be generated by different methods, imprinting the AB phase in parallel AB devices.

## 6 Qudits from enconded states by quantum transfered AB phases

Due the quantum nature of the system discussed above, the superposition principle will allow superpositions of the form , as qubits and qudit states. In a general form, the encoded states can be superposed in qudit states

(34) |

where and are functions of type .

This allow the quantum manipulation of these states in a higher hierarchy level. In particular, we have a set of operators that can leave these qudits invariant (as the case of an unitary evolution) or implement a quantum gate operation,

(35) |

One particular possible application in this context is the use in quantum memories [37, 38]. The interchanging between spintronic and optical scenarios by quantum transfer lead to a more consistent integrability among optical and solid state devices in quantum circuits with mixed devices.

## 7 One-qubit state modulated by AB phase

We can also use the quantum transfered AB phase states to codify a qubit of information in eq.(20). If after interaction the spin polarization is detected in the state , the non-classical state is projected in a superposition of coherent states carrying the AB phase . On the other hand, if the measurement is realized in the spin polarization , a superposition of coherent states carrying the AB phase terms is achieved. Taking the state eq.(20) in the normalized form

(36) |

where is a normalization factor, the arbitrary coefficients are achieved

(37) |

where and the coefficients depend on the AB phase

(38) |

This imply that the qubit can be appropriatelly modulated by the AB phase in the quantum transfer.

We can also rewrite the state

(39) |

We can incorporate the term in the normalization , such that we now have

(40) |

where

(41) |

Using the relation for

(42) |

that can also be written .

The state can then be put in the form

(43) | |||||

The projection in the state will lead to

(44) |

Setting the time of interaction , adjusting the high-Q cavity parameters, we achieve an AB phase dependent pattern (figure 2)

(45) |

The divergence is in fact the result of

## 8 Two-qubit spin-optical state modulated by AB phase

The state eq. (15) can be rewritten with the time of interaction in , adjusting the high-Q cavity parameters, such that we now have a two qubit state

(46) | |||||

where the coefficients satisfy the normalization condition

(47) |

and are given explicitly by

(48) | |||||

(49) |

and the normalization factor is

(50) |

We can in fact rewrite the state in the previous form

that simplifies to the state generated by dispersive interaction

(52) | |||||

It is important to note that action of an annihilation operators still leave the state in a two-qubit state

(53) |

where now the coefficients are -dependent

(54) | |||||

where the coefficients satisfy the condition

(55) |

As a consequence, given a coherent state, the AB-phase determines this two-qubit in an hypersphere of radius . Obviously, normalization can be obtained with the coefficients

(56) | |||||

(57) | |||||

(58) | |||||

(59) |

Another aspect to be emphasized is that the removal of photons by means of the action of annihilation operators can be done without properly destroy the two-qubit state, as what could be the case if the states where just number states.

We can also analyze the action of a bit-flip operator , that will keep the state as a two-qubit state

(60) |

that can also be written by

(61) | |||||

where now

(62) | |||||

(63) | |||||

(64) | |||||

(65) |

It is interesting to consider simultaneous action of the operators in (53) and (60). As a consequence, the operator

(66) |

is a parity operator in the two-qubit spin-optical state

(67) |

## 9 Spin-optical density operator modulated by AB phase, dissipation and Lindblad equation

We now consider the effect of a thermal bath in the density matrix carrying an AB-phase , corresponding to the state . We start with the dynamics of the density matrix coupled of the system in a thermal bath described by the following Lindblad equation

(68) |

where , and is a Lindblad superoperator.

We also consider the environment effect directly in the state as a result of the action of operators

(69) | |||||

Taking into account (15) and the operators encapsulating environment effect, we propose the ansatz

(70) | |||||

(71) |

where, for simplicity, a symmetric dissipation effect is considered in the scalar term

(72) |

is a time dependent function from the environment and is a coefficient of dissipation to the thermal bath. The operators are rewritten as follows

(73) | |||||

(74) |

Under explicit action of (73) and (74) on the coherent state, these operators are related to spin flipping operators

(75) | |||||

The density operator evolves according to the time evolution of these operators

(77) | |||||