Schwinger effect of a relativistic boson entangled with a qubit

# Schwinger effect of a relativistic boson entangled with a qubit

Yujie Li Collaborative Innovation Center of Advanced Microstructures, Fudan University,
Shanghai 200433, China
Qingqing Mao Collaborative Innovation Center of Advanced Microstructures, Fudan University,
Shanghai 200433, China
Yu Shi,111Corresponding author. Collaborative Innovation Center of Advanced Microstructures, Fudan University,
Shanghai 200433, China
###### Abstract

We use the concept of quantum entanglement to analyze the Schwinger effect on an entangled state of a qubit and a bosonic mode coupled with the electric field. As a consequence of the Schwinger production of particle-antiparticle pairs, the electric field decreases both the correlation and the entanglement between the qubit and the particle mode. This work exposes a profound difference between bosons and fermions. In the bosonic case, entanglement between the qubit and the antiparticle mode cannot be caused by the Schwinger effect on the preexisting entanglement between the qubit and the particle mode, but correlation can.

institutetext: Department of Physics & State Key Laboratory of Surface Physics, Fudan University,
Shanghai 200433, China
Department of Physics & State Key Laboratory of Surface Physics, Fudan University,
Shanghai 200433, China

## 1 Introduction

In relativistic quantum field theory, a particle is not eternal. Known as Schwinger effect, in a strong electromagnetic field, the vacuum decays into particle-antiparticle pairs Schwinger (), and likewise, a particle becomes a superposition state involving both particles and antiparticles. Many experimental efforts have been made to observe Schwinger effect rmp (), though have not yet been successful, because the rate is very low.

In this paper, the question we address is how the correlation and entanglement between a qubit and a bosonic particle is inherited by that between the qubit on one hand, and the particles and antiparticles generated by the Schwinger effect on the other. Here the qubit is a simple representation of another particle uncoupled with the electric field.

Recent years witnessed the application of the concepts and measures of quantum entanglement to various areas of quantum sciences. The measures of quantum entanglement can well characterize quantum correlations in quantum states and are independent of any observable. In field theory, we partition the system in terms of the modes shi (). Quantum entanglement in the Schwinger effect of Dirac or Klein-Gordon field, between a subsystem and the rest of the system, as measured by the von Neumann entropy of the reduced density matrix, was calculated Ebadi (); gavrilov (). Pairwise correlation and entanglement were also studied for Dirac field lidaishi (), by using mutual information and logarithmic negativity as the measures.

Pairwise correlation and entanglement are between two parts A and B, the combination of which is described in terms of the density matrix , obtained by tracing out other parts sharing a pure state. usually represents a mixed state, with pure state a special case. The reduced density matrix of A is , similarly, . The mutual information in is then nielsen ()

 I(ρAB)=S(ρA)+S(ρB)−S(ρAB), (1)

where is the von Neumann entropy of . if is a product pure state. Hence measures a kind of distance from a product pure state, containing both quantum entanglement and classical correlation. The logarithmic negativity , which is a measure of the quantum entanglement between A and B in , is defined as vidal ()

 N(ρAB)≡log2∥ρTAAB∥, (2)

where is the sum of the absolute values of the eigenvalues of the partial transpose of the original density matrix with respect to subsystem A. The partial transpose can also be made with respect to B, without changing the result of .

In this paper, we consider the Schwinger effect of a one-boson state, which transforms the one-boson state to a superposition of different number states of the particle and antiparticle modes. We study pairwise mutual information and quantum entanglement between a qubit and the particle mode, and that between the qubit and the antiparticle mode. An introduction to Schwinger effect is made in Sec. 2. The quantum state is described in Sec. 3. The correlation between the qubit and the particle mode is calculated in Sec. 4. The correlation between the qubit and the antiparticle mode is calculated in Sec. 5. Then the effect of a pulsed electric field is discussed in Sec. 6. A summary is made in Sec. 7.

## 2 Schwinger effect in a constant electric field

Consider a scalar field describing the bosons of mass and charge , coupled with an electric field along direction, satisfying the Klein-Gordon equation in the four-dimensional Minkowski space

 [(∂μ−ieAμ)(∂μ−ieAμ)+m2]ϕ(t,x)=0, (3)

where , is the scalar field, which can be expanded in terms of the mode functions as where denotes the momentum, is the annihilation operator of the particle, while is the creation operator of the antiparticle.

The Bogoliubov transformation between the in and the out modes, for and respectively, is

 ϕink=αkϕoutk+βkϕout∗−k, (4)

where and are Bogoliubov coefficients brout (); kim (); Ebadi ()

 αk=√2πΓ(−ν)e−iπ(ν+1)2,  βk=e−iπν, (5)

with , , satisfying . The corresponding annihilation and creation operators of the in and the out modes are related as

 aink=α∗kaoutk−β∗kbout†−k, (6)
 bink=α∗kboutk−β∗kaout†−k. (7)

Consequently the in-vacuum state for each mode becomes a superposition state of the out modes brout (); Ebadi (),

 (8)

where is the number of particles, is the number of antiparticles. It indicates the distribution of the created particles and antiparticles due to the Schwinger effect when an electric field is applied.

Similarly, from , one obtains

 (9)

which indicates the distribution of the created particles and antiparticles resulting from the effect of the electric field on one-particle state. We refer to this also as the Schwinger effect.

## 3 The initial entangled state

Now we investigate the influence of the electric field on the state of a qubit entangled with a bosonic particle of momentum , which is an excitation of the scalar field discussed above,

 |Φσ,q⟩=ε|↑⟩|0q⟩in+√1−ε2|↓⟩|1q⟩in, (10)

where is a coefficient, the basis states of the qubit are denoted as and . Obviously, the von Neumann entropy of the reduced matrices and are both equal to

 S(ε)=−ε2log2ε2−(1−ε2)log2(1−ε2). (11)

Being a pure state, the von Neumann entropy of is , therefore the mutual information

 I(Φσ,q)=2S(ε). (12)

The entanglement entropy, characterizing the entanglement between the qubit and the in mode is just .

With the mode also considered, can be rewritten as

 |Φσ,q,−q⟩in=(ε|↑⟩in|0q⟩in+√1−ε2|↓⟩in|1q⟩in)|0−q⟩in. (13)

Because of Bogoliubov transformation given in Eq.(8), one obtains

 |Φσ,q,−q⟩in =εαq∞∑n=0β∗nqα∗nq|↑,nq,n−q⟩out+√1−ε2|αq|2∞∑n=0β∗nqα∗nq√n+1|↓,(n+1)q,n−q⟩out. (14)

The density matrix is thus

 ρσ,q,−q =ε2|αq|2∞∑n,m=0β∗nqβmqα∗nqαmq|↑,nq,n−q⟩⟨↑,mq,m−q| +ε√1−ε2|αq|2αq∞∑n,m=0β∗nqβmqα∗nqαmq√m+1|↑,nq,n−q⟩⟨↓,(m+1)q,m−q| +ε√1−ε2|αq|2α∗q∞∑n,m=0β∗nqβmqα∗nqαmq√n+1|↓,(n+1)q,n−q⟩⟨↑,mq,m−q| (15)

which indicates that the Bogoliubov transformation causes in mode to be replaced by the out modes and . How the original correlation and entanglement are inherited between the qubit and these out modes will be investigated below. For brevity, we have omitted the superscript “out”.

## 4 Correlation and entanglement between the qubit σ and the mode q

We first study the correlation and entanglement between the qubit and the out mode . Tracing out the mode , we obtain the reduced density matrix of the qubit and , as

 ρσ,q= ε2|αq|2∞∑n=0∣∣∣βqαq∣∣∣2n|↑,nq⟩⟨↑,nq| +ε√1−ε2|αq|2αq∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↑,nq⟩⟨↓,(n+1)q| +ε√1−ε2|αq|2α∗q∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↓,(n+1)q⟩⟨↑,nq| +1−ε2|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n(n+1)|↓,(n+1)q⟩⟨↓,(n+1)q|. (16)

With the above summation expression, in the subspace of , , is a block matrix, with non-zero eigenvalues

 1|αq|2∣∣∣βqαq∣∣∣2n[ε2+(n+1)(1−ε2)|αq|2]. (17)

Tracing out the field mode in yields , which is

 ρσ=ε2|↑⟩⟨↑|+(1−ε2)|↓⟩⟨↓|, (18)

with eigenvalues , . This remains unchanged from the reduced density matrix of qubit obtained from in (10), as nothing is done on the qubit .

Tracing out the qubit in yields , which is

 ρq=1|αq|2∞∑n=0∣∣∣βqαq∣∣∣2n[ε2+n(1−ε2)|βq|2]|nq⟩⟨nq|, (19)

with eigenvalues

 1|αq|2∣∣∣βqαq∣∣∣2n[ε2+n(1−ε2)|βq|2],  n=0,1,2,⋯⋯ (20)

Then we obtain the mutual information as

 I(ρσ,q) =−ε2log2ε2−(1−ε2)log2(1−ε2) −∞∑n=0|βq|2n|αq|2(n+1)[ε2+n(1−ε2)|βq|2]log2[|βq|2n|αq|2(n+1)(ε2+n(1−ε2)|βq|2)] +∞∑n=0|βq|2n|αq|2(n+1)[ε2+(n+1)(1−ε2)|αq|2]log2[|βq|2n|αq|2(n+1)(ε2+(n+1)(1−ε2)|αq|2)], (21)

which depends on the coefficient parameter and the strength of the electric field . When , reduces to .

The dependence of the mutual information on the electric field and the parameter is shown in Fig. 1. For a fixed value of , monotonically decreases with the increase of the electric field and asymptotically approaches a certain nonvanishing value independent of . For less or larger than , the closer to the parameter is, the quicker decreases with the increase of when is small. For any given value of and for less or larger than , the farther to the parameter is, the smaller is. The mutual information becomes zero as or , in which case the mutual information vanishes even in the absence of the electric field.

We use logarithmic negativity to measure the entanglement. After making the partial transpose of the density matrix with respect to , we obtain

 ρTσσ,q= ε2|αq|2∞∑n=0∣∣∣βqαq∣∣∣2n|↑,nq⟩⟨↑,nq| +ε√1−ε2|αq|2αq∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↓,nq⟩⟨↑,(n+1)q| +ε√1−ε2|αq|2α∗q∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↑,(n+1)q⟩⟨↓,nq| +1−ε2|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n(n+1)|↓,(n+1)q⟩⟨↓,(n+1)q|, (22)

which is a block matrix in the subspace of , . Therefore the eigenvalues of are

 ε2|αq|2,
 12|αq|2∣∣∣βqαq∣∣∣2n[ ∣∣∣βqαq∣∣∣2ε2+n(1−ε2)|βq|2± (23)

Thus the logarithmic negativity is

 N(ρσ,q) =log2⎡⎢⎣ε2|αq|2+∞∑n=0|βq|2n|αq|2(n+1) ⎷(∣∣∣βqαq∣∣∣2ε2+n(1−ε2)|βq|2)2+4ε2(1−ε2)|αq|2 ⎤⎥⎦, (24)

which describes the quantum entanglement between and . when .

Fig. 2 shows how the logarithmic negativity depends on the strength of the electric field and the parameter . The variation trend of with respect to and is similar to that of . But when is small, decreases more rapidly with the increase of than does, indicating that the entanglement is more sensitive to the coupling with the electric field. Like , monotonically decreases with the increase of the electric field and approaches a certain nonzero asymptotic value as .

The Schwinger effect of two entangled fermions of momenta and  lidaishi () can reduce effectively to a fermion counterpart of our present bosonic problem, under the constraint that the electric field does not couple the mode , thus fixing the Bogoliubov coefficients of the mode to be , , thereby reducing mode to our qubit . With the increase of , the monotonic decrease towards of the mutual information and entanglement between fermionic mode and mode reduces to those between the qubit and fermionic mode here. The boson-fermion comparison will be discussed in the summary.

It is also interesting to make comparison with the bosons in Unruh effect fuentes1 (); Eduardo (); Pan () and near a dilaton black hole Jieci (), with the role of the electric field in our case replaced as the acceleration, but the Bogoliubov coefficient that is the counterpart of can be arbitrarily large, making the entanglement disappear in the limiting cases. In contrast, in our present case, , consequently the entanglement in persists as .

## 5 Correlation and entanglement between the qubit σ and the mode −q

Now we study the correlation and the entanglement between and . Tracing out the mode , we obtain the reduced density matrix of and , , as

 ρσ,−q= ε2|αq|2∞∑n=0∣∣∣βqαq∣∣∣2n|↑,n−q⟩⟨↑,n−q| +ε√1−ε2β∗q|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↑,(n+1)−q⟩⟨↓,n−q| +ε√1−ε2βq|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↓,n−q⟩⟨↑,(n+1)−q| +1−ε2|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n(n+1)|↓,n−q⟩⟨↓,n−q|, (25)

which is a block matrix in the subspace of , , thus the non-zero eigenvalues of are

 1|αq|2∣∣∣βqαq∣∣∣2n[ε2+n(1−ε2)|βq|2],  n=0,1,2,⋯. (26)

Tracing out the field mode in yields , which is

 ρσ=ε2|0p⟩⟨0σ|+(1−ε2)|1p⟩⟨1σ|, (27)

with eigenvalues , .

Tracing out in yields , which is

 (28)

with eigenvalues

 1|αq|2∣∣∣βqαq∣∣∣2n[ε2+(n+1)(1−ε2)|αq|2],  n=0,1,2,⋯. (29)

According to the definition of the mutual information between modes and , we have

 I(ρσ,−q) =−ε2log2ε2−(1−ε2)log2(1−ε2) +∞∑n=0|βq|2n|αq|2(n+1)[ε2+n(1−ε2)|βq|2]log2[|βq|2n|αq|2(n+1)(ε2+n(1−ε2)|βq|2)] −∞∑n=0|βq|2n|αq|2(n+1)[ε2+(n+1)(1−ε2)|αq|2]log2[|βq|2n|αq|2(n+1)(ε2+(n+1)(1−ε2)|αq|2)]. (30)

The dependence of the mutual information on the electric field and the parameter is shown in Fig. 3. monotonically increases with the increase of the electric field , and asymptotically approaches a certain value independent of . For larger or smaller than , the closer to the parameter is, the larger the asymptotic value is. Moreover, when is small, for larger or smaller than , the closer to the parameter is, the quicker increases with the increase of , and the larger the value of is.

Now we calculate the logarithmic negativity of . After making the partial transpose of the density matrix with respect to the mode , one obtains

 ρTσσ,−q= ε2|αq|2∞∑n=0∣∣∣βqαq∣∣∣2n|↑,n−q⟩⟨↑,n−q| +ε√1−ε2β∗q|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↓,(n+1)−q⟩⟨↑,n−q| +ε√1−ε2βq|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n√n+1|↑,n−q⟩⟨↓,(n+1)−q| +1−ε2|αq|4∞∑n=0∣∣∣βqαq∣∣∣2n(n+1)|↓,n−q⟩⟨↓,n−q|, (31)

which is a block matrix in the subspace of , with eigenvalues

 1−ε2|αq|4,
 12|αq|2∣∣∣βqαq∣∣∣2n[ ε2+(n+2)(1−ε2)|βq|2|αq|4±   ⎷(ε2+(n+2)(1−ε2)|βq|2|αq|4)2−4ε2(1−ε2)|βq|2|αq|4 ⎤⎥ ⎥⎦,  n=0,1,2,⋯. (32)

Thus the logarithmic negativity of is

 N(ρσ,−q) =log2[1−ε2|αq|4+∞∑n=0|βq|2n|αq|2(n+1)(ε2+(n+2)(1−ε2)|βq|2|αq|4)] =log21=0, (33)

which means starting from the initial state (13 ) with any value of the parameter , which is entangled between and in mode , with the action of the electric field, the entanglement between and out mode always vanishes.

From the expressions of the mutual information of and , we obtain

 I(ρσ,q)+I(ρσ,−q)=−2[ε2log2ε2+(1−ε2)log2(1−ε2)], (34)

implying that Schwinger effect redistributes the total correlation in the initial entangled state, into and . However, there is no such an identity for the logarithmic negativity, hence Schwinger effect does not redistribute quantum entanglement. The reason is that the qubit is not coupled with the electric field. In the fermion model studied previously lidaishi (), even when reduced to the problem of an uncoupled qubit and the fermionic mode coupled with the electric field, redistribution exists both in mutual information and in logarithmic negativity. See Eqs. (38-39) in Ref.  lidaishi (), where the fermionic mode coupled with the electric field is denoted as , and the uncoupled mode is equivalent to a qubit.

## 6 Effect of a pulsed electric field

Now we investigate the effect of a pulsed electric field. Consider a Sauter-type electric field along direction, where is the width of the pulsed electric field sauter (). The gauge potential can be chosen as

 Aμ=(0,0,0,−E0τtanh(tτ)), (35)

for which the Bogoliubov transformation yields kim (); Ebadi ()

 |αk|2=cosh[πτ(ωoutk+ωink)]+cosh(2πλ)2sinh(πτωink)sinh (πτωoutk), (36)
 |βk|2=cosh[πτ(ωoutk−ωink)]+cosh(2πλ)2sinh(πτωink)sinh (πτωoutk), (37)

where

 λ=√(eE0τ2)2−14, (38)
 ωink=√(kz+eE0τ)2+k2⊥+m2, (39)
 ωoutk=√(kz−eE0τ)2+k2⊥+m2. (40)

As , , then and , reducing the problem to the case without the electric field. As , , which means and reduce to the values in the case of the constant electric field.

The analyses and calculations for and above for the case of a constant electric field can be applied to the pulsed electric field, but with and now given in Eqs. (36) and (37). Hence the mutual information and logarithmic negativity now depend on not only but also .

In parallel with the above study on a constant electric field, we first investigate the influence of a pulsed electric field on the entanglement and correlation between and mode .

The influence of the pulsed electric field on the mutual information is shown in Fig. 4, which indicates its dependence on the strength and the width of the pulsed electric field. For smaller than a certain value, with the increase of , monotonically decreases and approaches the asymptotic value dependent on as . For larger than a certain value, with the increase of , first decreases to a minimum and then increases to a maximum before finally decreases and approaches asymptotically a value dependent on as . The larger is, the smaller the values of corresponding to the minimum and the maximum of . When is small, with the increase of , decreases to a minimum and then increases to a maximum, and finally decreases and approaches asymptotically a certain value independent of . For a given value of , the variation trend of with respect to is opposite to that of with respect to for a given value of . Moreover, When is smaller than a certain value, the larger the values of corresponding to the minimum and the maximum of . When is larger than a certain value, decreases monotonically with the increase of and asymptotically approaches a value independent of . As , the dependence of on is the same as that the case of constant electric field for , as shown in Fig. 1.