Emission of time-bin entangled particles

This section is closely based on Ref. [Hofer13].

Here we investigate the mesoscopic capacitor in a quantum spin Hall insulator. We note that Inhofer and Bercioux independently presented a similar analysis [Inhofer13]. Because this system makes use of helical edge state, we denote it as thehelical capacitor.

Figure 2.5: Mesoscopic capacitor in a quantum spin Hall insulator. An elongated pair of helical edge states is coupled via a QPC to a quantum dot. A top gate potential U(t) is used to move energy levels above and below the Fermi energy resulting in the emission of electrons and holes. A magnetic flux φthrough the dot breaks time-reversal symmetry and allows for backscattering to occur. Figure reprinted from Ref. [Hofer13].

The helical capacitor is sketched in Fig. 2.5. Since there are two counter-propagating edge states, the scattering matrix is a two-by-two matrix. As long as time-reversal symmetry is preserved, backscattering between the edge states is forbidden. As we show below, the helical capacitor is then equivalent to two copies of the chiral capacitor. A more interesting situation occurs if time-reversal symmetry is broken by a magnetic flux φwhich pierces the quantum dot.

Current emitted by the helical capacitor

As in Ref. [Delplace12], we assume the region of the QPC to be too small to be affected by the magnetic flux. Scattering at the QPC is then given by a four-by-four scattering matrix (two spin species at either side of the QPC) which fulfills the constraint

S_{QP C} = ΘS_{QP C}^† Θ⁻¹, (2.56)

where Θ denotes time reversal. This condition constrains scattering at the QPC such that it can be described by a reflection amplitude r, a spin-preserving transmission amplitude d and a spin-flipping transmission amplitudedσ. Here we choose these amplitudes to be real as their phases have no effect on the observables under consideration. We note that although spin flipping processes within a pair of helical edge states is forbidden, the spin can flip when tunneling between the elongated pair of states and the pair of states within the dot.

The frozen scattering matrix for the helical capacitor can then be found by considering all possible paths an electron can take and reads [Delplace12;Hofer13]

S(E, t) =

r−d²Z₋−d²_σZ₊ dd_σZ¯ dd_σZ¯ r−d²Z₊−d²_σZ₋

, (2.57)

where

Z_±= ei2π[E−eU(t)]/∆±iφ

1−rei2π[E−eU(t)]/∆±iφ, (2.58)

Z¯=Z₊−Z₋ and ∆ the level spacing in the dot. The phase picked up during one round trip in the quantum dot consists of the Aharonov-Bohm phase φ and the dynamical phase which is equivalent to the chiral case [cf. (2.44)]. The basis of the scattering matrix is chosen as (R, L), whereR (L) denotes right-movers (left-movers) and transmission amplitudes are on the diagonals.

In the absence of a magnetic flux, the off-diagonal terms of the scattering matrix vanish indicating the absence of backscattering. The diagonal terms become equal and correspond to a chiral capacitor with QPC transparency

D=d²+d²_σ= 1−r². (2.59)

We are thus left with two identical but uncoupled copies of the chiral capacitor.

A magnetic flux has the effect of breaking the degeneracy in the quantum dot leading to different emission times for the right- and left-movers. However, if eitherdordσ vanishes, backscattering is again suppressed and the helical capacitor is equivalent to two uncoupled chiral capacitors. In this case however, their emission times might differ due to the magnetic flux.

In the most general case, Eq. (2.57) can be diagonalized through the transformation S_d(E, t) =V^†S(E, t)V =

In the eigenbasis, the helical capacitor can thus still be described by two independent chiral capacitors [cf. Eq. (2.44)]. In this section we focus on the adiabatic limit, where we can approximate

r−DZ±≈ t−t^E_±+iΓ^E_±

t−t^E_±−iΓ^E_±, (2.61)

where the timest^E_± are defined through the equations

eU(t^E_±) =E±∆φ/2π, (2.62)

In the eigenchannels of the scattering matrix the current is thus given by Eq. (2.49) with the substitution t^EF →t_±=t^E_±^F and Γ→Γ_±= Γ^E_±^F. Transforming back to the basis of right- and left-movers, we find that around the time a right-mover crosses the Fermi energy, the current reads

I(t) =

where I_R(L) denotes the current in the right-moving (moving) channel. The current emitted by a left-mover crossing the Fermi energy is obtained by the substitution d↔d_σ and − →+. The current and the energy levels for a top gate potential that moves one (split) Kramers pair above and below E_F is illustrated in Fig.2.6.

Due to the magnetic flux, the energy levels corresponding to right- and left-movers (or different spins) are no longer degenerate and cross the Fermi energy at different times. If bothdanddσ are finite, the particles will be emitted in a superposition of right- and left-movers. The relative height of the current peaks in the two outgoing channels is solely determined by the ratio ofd²tod²_σ [cf. (2.64)] and the integral over the sum of the currents (IR+IL) corresponds to exactly one electron (hole) per dip (peak).

By locally breaking time-reversal symmetry, the helical capacitor thus becomes a proper single-particle source emitting one particle after the other. Allowing for spin flips in the emission process, the emitted particles are in a superposition of right- and left-movers. As we discuss below, this can lead to the emission of time-bin entangled particles. Before turning to the produced entanglement however, we further characterize the helical capacitor by its zero frequency noise.

Noise produced by the helical capacitor

Considering a top gate potential which moves one Kramers pair above and below the Fermi energy within one period, we evaluate Eq. (2.19) which yields the autocorrelated zero frequency noise (we omit the channel

Figure 2.6: Energy levels and current emitted by the helical capacitor. (a) Energy levels in the quantum dot as a function of time (Eldenotes the levels in the absence of a top gate potential). The time-dependent top gate potentialU(t) =U₀+U₁cos(Ωt) moves one split Kramers pair above and below the Fermi energy.

Att₋ (t₊), a level corresponding to a right-mover (left-mover) crossesE_F. (b) This leads to the emission of an electron at t₋ and t₊ resulting in a negative current pulse of Lorentzian shape. If the electrons can flip their spin when leaving the quantum dot, the emitted particles are emitted in a superposition of right- and left-movers. Parameters: eU₀=−∆/8,eU₁= ∆/4,d²= 0.15,d²_σ= 0.05,φ=π/6.

indices because we haveP =P^RR=P^LL =−P^RL =−P^LR) P =P^e+P^h, Pⁱ= e²Ω

2π 4d²d²_σ

D² (

1− 4Γⁱ₋Γⁱ₊ Γⁱ₊+ Γⁱ₋²

+ ¯t²_i )

. (2.65)

Here i =e/h labels electron or hole emission and ¯t_i =tⁱ₊−tⁱ₋. The noise is a result of the uncertainty of how many particles are emitted into each channel during one period. Consequently, it vanishes if either d, dσ or φ vanish. In all those cases, there is exactly one electron and one hole emitted into each outgoing channel during one period. Note that in the time-reversal symmetric case (φ= 0), we have Γⁱ₋ = Γⁱ₊ and t¯ⁱ= 0 both for electron as well as hole emission.

The last expression is equivalent to the noise generated by two chiral capacitors which have their outgo-ing channels coupled by a QPC with transmission and reflection probabilities d²/D, d²_σ/D in a Hong-Ou-Mandel (HOM) setup [Hong87; Liu98; Burkard00; Giovannetti06; Jonckheere12]. This setup was proposed by Ol’khovskaya et al. [Ol’khovskaya08] and experimentally realized by Bocquillon et al. [Bocquillon13].

Whereas in those works two chiral capacitors emit one-particle each toward a central QPC with a difference, in our case a single helical capacitor emits two particles through the same QPC with a time-difference ¯t determined by the magnetic flux.

If the top gate potential is linear in time throughout the emission of Kramers pairs, i.e. Γⁱ₋ = Γⁱ₊, the noise can be expressed solely as a function of the magnetic fluxφand the transmission probabilities

P^e=P^h=e²Ω 2π

4d²d²_σ D²

φ²

(D/2)²+φ². (2.66)

Note that the electron and hole emission can happen with different velocities (i.e. Γ^e_± 6= Γ^h_±). The last expression is plotted in Fig. 2.7.

Emission of time-bin entangled particles

To investigate the generated entanglement, we consider the outgoing state upon the emission of one Kramers pair noting that in analogy to the zero frequency noise, electron and hole emission can be treated separately.

At zero temperature, the mesoscopic capacitor will create two single-particle excitations above the undis-turbed Fermi sea. In terms of the operators describing outgoing scattering states in the eigenbasis of the scattering matrix, the state reads [Keeling06;Keeling08]

|Ψouti= ˆB₊^†Bˆ₋^†|0i, Bˆ_±=√ 2ΓX

E>0

e^{E(ite±−Γe±)}ˆb_±(E). (2.67)

0 0 0.2 0.4 0.6 0.8

1 0 -1 -2

-30 20 40 60 80 100

Figure 2.7: Zero temperature shot noise created by the source as a function of the magnetic flux for a linear top gate potential. At zero flux, equality of the emission times forces the particles into different outputs. For finite flux, the emission times differ and the noise increases until it saturates when the emitted particles have no overlap anymore. The noise is proportional to the concurrence created per cycle. The inset shows the excess noise divided by the zero temperature noise as a function of temperature. Here: d²= 0.15,d²_σ= 0.05 and 2πΩΓ = 0.1 in accordance with the adiabatic limit. Figure taken from Ref. [Hofer13].

Here |0i denotes the Fermi sea and ˆb_±(E) annihilates an outgoing state in the scattering eigenchannel denoted by the index ±[cf. Eq. (2.60)]. Note that this expression only holds in the adiabatic limit. When transforming the last expression to the basis of right- and left-movers, the state will consist of terms which correspond to both particles being emitted into the same channel in addition to the terms where both a right-and a left-mover are emitted. Here we are interested in the entanglement between the spatially separated parties which receive the right- and left-movers respectively. We discard events where one party receives both particles by post-selection which is equivalent to projecting on the subspace where one electron ends up with either party. We note that this post-selection is a local operation that the parties can perform without communication and thus can not create any entanglement between the two parties [Bennett96] as discussed in Sec. 2.2. The (unnormalized) relevant state then reads

|Ψ_LRi= 1

D d²|+_L,−^Ri+d²_σ|−^L,+_Ri

, (2.68)

where |±^L/Ri = ˆB_L/R±^† |0i describes an electron emitted at time t_± into channel L/R and the operator Bˆ_L/R± is obtained by substituting ˆb±→ˆbL/R in Eq. (2.67). The normalization of Eq. (2.68) is chosen such that the absolute square gives the probability of finding an electron in each channel. We note that the above basis states are not orthogonal. Indeed we find

|h−^α|+βi|²=δα,β|h−|+i|²=δα,β

4Γ^e₋Γ^e₊ Γ^e₊+ Γ^e₋2

+ ¯t²_e

, (2.69)

where α, β=L, R. Using this expression, we can write the zero frequency noise as P^e=e²Ω

2π 4d²d²_σ

D² 1− |h−|+i|²

, (2.70)

where we omitted the channel indices. The last expression corresponds to the noise describing a HOM experiment with Fermions in the states |+i and |−i[Bocquillon13] and directly connects the noise to the wavefunctions of the emitted particles.

Ift^e₊6=t^e₋, the state in Eq. (2.68) is entangled in the emission times. In case a right-mover is emitted at t^e₊, a left-mover has to be emitted att^e₋ and vice versa. This entanglement in temporal modes is known as time-bin entanglement [Brendel99]. In general, the entanglement can be characterized by the concurrence produced per cycle [Wootters98; Beenakker05]

C=|hΨ_LR|σˆ^L_y ⊗σˆ^R_y|Ψ_LRi|, (2.71)

where the Pauli matrices act on the entangled degree of freedom. In order to define the action of the Pauli matrices in the last expression, we write

|+αi=h−|+i|−^αi+p

1− |h−|+i|²|×^αi. (2.72) The states |−^αi and ×^αi then constitute an orthonormal basis and we can define the action of the Pauli matrices in Eq. (2.71) as

σ_y^α|−^αi=i|×^αi, σ^α_y|×^αi=−i|−^αi. (2.73) Using the last two equations, we find for the concurrence produced per cycle

C= 2d²d²_σ

D² 1− |h−|+i|²

. (2.74)

As expected, the entanglement vanishes when the two emission times coincide (i.e. |+i=|−i). It reaches a maximum of C = 1/2 in the case of well separated emission times if d² = d²_σ. In this case the source emits Bell states with probability one half. This coincides with the maximal entanglement produced by dynamical electron-hole entanglers [Samuelsson05a; Beenakker05]. In the case of well separated emission times, the emitted electrons are uncorrelated implying that there is no entanglement between the two elec-trons. However, each electron is in a superposition of left- and right-mover. As discussed in Secs. 2.2and 2.4.2, each electron is thus in an entangled state with respect to the bi-partition of left- and right-movers.

The post-selection (which is a local operation) then maps this single-particle entanglement onto two-particle entanglement [cf. Eqs. (2.67) and (2.68)].

Interestingly, the concurrence is proportional to the zero frequency noise (π/e²Ω)P^e=C. We note that this is not a fundamental relation but a consequence of the mechanism that generates the entanglement.

Whereas the zero frequency noise is a consequence of the uncertainty in the number of particles emitted in either channel, the concurrence arises from the part of the state where one particle is emitted into either channel.

Finite temperatures

Finally, we investigate the influence of finite temperature. Here we consider temperatures that are much smaller than the energy scale over which the scattering matrix changes. In our case this is determined by the level width given in Eq. (2.46). For a level spacing ∆≈2.5K (as in the experiment of Ref. [F`eve07]), this restricts our analysis to temperatures kBT 500mK. In this regime, only the noise is modified by the temperature. We define the excess noise P^T as the total noise minus the noise for U(t) = φ= 0, i.e.

in the absence of dynamics and backscattering. The excess noise turns out to be proportional to the zero temperature noise with the proportionality constant (here Γⁱ₊= Γⁱ₋ = Γ)

PT^e/P^e= 4Ω²Γ²

∞

X

q=1

qcoth qΩ

2kBT

e^−2qΩΓ−6k_BT

Ω ΩΓ. (2.75)

This expression only holds in the limit ΩΓ → 0 since we consider a single resonance level of the dot in the adiabatic limit. The last expression is plotted in the inset of Fig. 2.7. We note that the temperature independence of the shape of the noise also occurs in HOM experiments with levitons [Dubois13a;Dubois13b], which (for well separated pulses) are described by the same scattering matrix as the mesoscopic capacitor in the adiabatic limit.

In analogy to Ref. [Samuelsson09], we expect the entanglement to survive up to temperatures on the order of the frequency Ω. Realistic temperatures thus require the capacitor to be operated away from the adiabatic limit. Although the current and the noise depend on the exact driving potential, we expect the adiabatic limit to qualitatively capture the physics. In particular, while the wavefunction of the emitted particles depend on the driving potential, we expect Eqs. (2.70) and (2.74) (and therefore the proportionality of the noise and the concurrence) to hold even in the non-adiabatic regime.

Summary

We propose a single-electron source which emits particles into helical edge states. As long as time-reversal symmetry is preserved, the source is equivalent to two copies of the chiral capacitor. By locally breaking time-reversal symmetry, the source becomes a proper single-particle emitter. Allowing for spin flips in

the emission process, the emitted particles are in a superposition of left- and right-movers and the source becomes noisy. This superposition of spatially separated states can be used to create pairs of time-bin entangled particles. In principle, this entanglement can be used to violate a Bell inequality using a Franson interferometer [Franson89] in analogy to Ref. [Splettstoesser09]. The concurrence of the entangled particles is proportional to the zero frequency noise, making it experimentally accessible, and contains information on the emitted wavefunctions.