Deterministic quantum state transfer from an electronic charge qubit to a photonic polarization qubit

Deterministic quantum state transfer from an electronic charge qubit

to a photonic polarization qubit

Ament, L.J.P.; Beenakker, C.W.J.

Citation

Ament, L. J. P., & Beenakker, C. W. J. (2006). Deterministic quantum state transfer from an

electronic charge qubit to a photonic polarization qubit. Physical Review B, 73(12), 121307.

doi:10.1103/PhysRevB.73.121307

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Leiden University Non-exclusive license

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https://hdl.handle.net/1887/77076

Deterministic quantum state transfer from an electronic charge qubit

to a photonic polarization qubit

L. J. P. Ament and C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

共Received 7 February 2006; published 31 March 2006兲

Building on an earlier proposal for the production of polarization-entangled microwaves by means of intraband transitions in a pair of quantum dots, we show how this device can be used to transfer an unknown single-qubit state from electronic charge to photonic polarization degrees of freedom. No postselection is required, meaning that the quantum state transfer happens deterministically. Decoherence of the charge qubit causes a nonmonotonic decay of the fidelity of the transferred state with an increasing decoherence rate. DOI:10.1103/PhysRevB.73.121307 PACS number共s兲: 78.67.Hc, 03.65.Yz, 42.50.Dv, 78.70.Gq

Quantum state transfer between matter and light is a key step in the development of scalable quantum networks. A qubit ␣兩0典+␤兩1典 is encoded in matter degrees of freedom for the purpose of computation, and then transferred to pho-tonic degrees of freedom for transportation to a distant loca-tion共where it might be converted back to matter for storage or further processing兲. A promising scheme1 _{to accomplish}

this in the context of atomic physics uses a laser beam to transfer the internal state of an atom to the optical state of a cavity mode. The matter qubit in this case is a superposition of two degenerate ground states of the atom and the photonic qubit is the superposition of an occupied and an empty cavity mode. An all-electronic analog of this scheme, to transfer a state from one matter qubit to another, has been proposed as well.2

In the context of semiconductor quantum dots, there exist several proposals for the transfer of a quantum state from electron spin degrees of freedom共spin qubit兲 to photon po-larization degrees of freedom.3,4_{A separate line of}

investiga-tion in this context involves a charge qubit,5,6_{i.e., a}

single-electron state␣兩A典+␤兩B典 delocalized over a pair of quantum dots A and B. The coupling of a charge qubit to a photon cavity mode was investigated in Ref. 7, as a way to produce polarization-entangled photon pairs at microwave frequen-cies. We build on that proposal to show that the combination of a microwave resonator and three quantum dots can be used to transfer an arbitrary single-qubit state from electron charge to photon polarization degrees of freedom.

The device for the quantum state transfer, shown sche-matically in Fig. 1, differs from the photon entangler7_{only in}

that it produces a single photon rather than a photon pair. The resonant transitions involve a total of five electron levels in three quantum dots: three ground states A , B , C and two ex-cited states A

⬘

, B

⬘

. The radiative transitions A

⬘

↔A and

B

⬘

↔B are resonant with a cavity mode. As explained in

detail in Ref. 7, the confining potential and magnetic field can be arranged such that the transition in dot A couples only to the left circular polarization␴+and the transition in dot B

couples only to the right circular polarization␴−.

The charge qubit is prepared initially in the state, 兩⌿in典 = 共␣兩A

⬘

典 +␤兩B

⬘

典兲兩0, 共1兲

where兩0典 denotes the photon vacuum. This single-electron state in dots A and B can decay into a reservoir via a third dot

C, leaving behind a photon in the cavity. The quantum state

transfer has succeeded if the final state is

兩⌿final典 = 兩O典共␣兩 + 典 +␤兩− 典兲, 共2兲

where兩O典 is the electron vacuum 共all quantum dots empty兲 and兩±典 represents the two photon states of opposite circular polarization. For later use, we also define the states

兩⌿±典 =␣兩 + 典 ±␤兩− 典, 共3兲

兩⌽±典 = 共兩A典 ± 兩B典兲/

冑

2. 共4兲

The reversible radiative transitions 共with rate g兲 are de-scribed by the Hamiltonian

FIG. 1. Schematic of the model. The upper panel shows a top view of three quantum dots A , B , C connected to an electron reser-voir, the lower panel shows the resonant energy levels in the quan-tum dots. An electron can tunnel between dots A and C or between dots B and C共solid arrows兲, but not directly between dots A and B. From dot C, the electron can tunnel into the reservoir共dashed ar-row兲, while the reverse process is prevented by a large bias voltage. A radiative transition within dots A or B is accompanied by the emission or absorption of a photon, with, respectively, left共␴+兲 or

right 共␴−兲 circular polarization. The coupled electron-photon

dy-namics transfers the charge qubit␣兩A⬘典+␤兩B⬘典 to the photon qubit ␣兩 +典+␤兩−典.

PHYSICAL REVIEW B 73, 121307共R兲 共2006兲

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Hg= g共兩A + 典具A

⬘

0兩 + 兩B − 典具B

⬘

0兩兲 + H.c. 共5兲 The reversible tunnel transitions共with rate T兲 between dots A or B and dot C have the Hamiltonian

HT= T共兩C典具A兩 + 兩C典具B兩兲丢1photon+ H.c., 共6兲

where1photon is the unit operator acting on the photons.

The irreversible escape共with rate ⌫兲 of the electron from dot C into the reservoir is described by the jump operator

D_⌫=

冑

⌫ 兩O典具C兩丢1photon. 共7兲

These operators determine the time evolution of the density matrix␳共t兲 through the master equation8,9

d␳ dt = − i关H,␳兴 + D⌫␳D⌫ † −1 2共D⌫ † D_⌫␳+␳D_⌫†D_⌫兲. 共8兲

共We have set ប to 1.兲 The initial condition is ␳共0兲 =兩⌿in典具⌿in兩.

Inspection of the master equation shows that the density matrix evolves entirely in the five-dimensional subspace spanned by the states

兩u1典 = 兩⌿in典, 兩u2典 = 兩⌽+典兩⌿+典, 兩u3典 = 兩⌽−典兩⌿−典,

兩u4典 = 兩C典兩⌿+, 兩u5=兩⌿final典, 共9兲

of even parity under the exchange ␣↔␤, A↔B, A

⬘

↔B

⬘

, ␴+↔␴−. The states共␣兩A

⬘

典−␤兩B

⬘

典兲兩0典, 兩⌽+典兩⌿−典, 兩⌽−典兩⌿+典,

兩C典兩⌿−典, and 兩O典兩⌿−典 of odd parity do not appear.

The five-dimensional subspace may be further reduced to a four-dimensional subspace by noting that the master equa-tion共8兲 couples only to␳55and to␳ijwith i , j艋4. The matrix elements ␳ij with i = 5 , j⫽5 or j=5,i⫽5 remain zero. We may, therefore, seek a solution of the form

␳共t兲 =˜␳共t兲 + 关1 − Tr˜␳共t兲兴兩⌿_final典具⌿_final兩, 共10兲

where ␳˜ is restricted to the four-dimensional subspace

spanned by the states兩ui典 with i艋4. The evolution equation for˜ reads␳

d␳˜ dt = M˜ +␳ ˜ M␳ †_{, 共11兲} M = −

_冑

1 2

冢

0 ig ig 0 ig 0 0 2iT ig 0 0 0 0 2iT 0 ⌫/

冑

2

冣

. 共12兲 The solution ␳ ˜共t兲 = eMt_{˜␳共0兲e}M†t _共13兲 has a lengthy expression in terms of the eigenvalues and eigenvectors of the matrix M. What is important for the de-terministic quantum state transfer is that all four eigenvalues ␮i have a negative real part, for any nonzero g, T, and ⌫. This implies that˜␳共t兲→0 for t→⬁, so␳共t兲→兩⌿_final典具⌿_final兩.

The fidelity of the quantum state transfer,

F = 具⌿final兩␳兩⌿final典 = 1 − Tr˜␳ 共14兲

approaches unity in the long-time limit with a rate deter-mined by the eigenvalue with the real part closest to zero,

lim t_→⬁

1 −F共t兲 ⬀ e−␣t, ␣= 2 min共兩Re␮i兩兲. 共15兲 The asymptotic limits of␣ are

␣=

冦

T2⌫/2g2 for gⰇ T,⌫

g2⌫/8T2 for TⰇ g,⌫ 2T2/⌫ for ⌫ Ⰷ g,T

.

冧

共16兲

If one varies⌫ and T at fixed g, the rate␣reaches its maxi-mum of␣max= g at T / g = 1 ,⌫/g=4. 共See Fig. 2.兲 That␣

van-ishes for large⌫,T,g can be understood as a manifestation of the quantum Zeno effect:10 _{the electron remains trapped in}

the quantum dots because the decay into the reservoir is in-hibited by a too frequent measurement. The optimal rate ␣max= g implies that emission or absorption of the photon in

the cavity is needed to effectively transfer the state. This seems fast enough, in view of the inevitable losses in the cavity.

The fidelity of the quantum state transfer is reduced by decoherence of the charge qubit due to coupling of the charge to acoustic phonons11,12 _{or to the background charge}

fluctuations.13_{Following Ref. 14 we model this decoherence}

共with rate ⌫␾兲 by means of the jump operators, DX=

冑

⌫␾共兩X典具X兩 + 兩X

⬘

典具X

⬘

兩兲丢1photon,

X苸 兵A,B其, DC=

冑

⌫␾兩C具C兩丢1photon, 共17兲

which measure the charge on each of the three dots. The master equation共8兲 now becomes

d␳ dt = − i关H,␳兴 +X=⌫,A,B,C

兺

DX␳DX † −1 2_X=⌫,A,B,C

兺

共

DX † DX␳+␳DX † DX

兲

, 共18兲 where D_⌫ was defined in Eq. 共7兲. In what follows, we take

FIG. 2. Contour plot of the rate␣ 关defined in Eq. 共15兲兴, at which the fidelity of the quantum state transfer approaches unity, as a function of the tunnel rates T and⌫. All rates are normalized by the electron-photon coupling constant g.

L. J. P. AMENT AND C. W. J. BEENAKKER PHYSICAL REVIEW B 73, 121307共R兲 共2006兲

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␣=␤=1₂

冑

2: since the initial state共1兲 is then maximally de-localized, it will be most sensitive to decoherence.

Results forF_⬁= limt_→⬁F共t兲 are plotted in Fig. 3 for two parameter choices. The asymptotic limits are

F_⬁=

再

1 − A⌫1 ␾ for ⌫␾Ⰶ ⌫,T,g 2+ B⌫␾ −4 for ⌫_␾Ⰷ ⌫,T,g,

冎

共19兲 A =4g 4_{+ 16T}4_{+ g}2_⌫2_{− 10g}2_T2 4g2T2⌫ , 共20兲 B = T2_共3g2_{− 2T}2_{兲. 共21兲}

Note that B is independent of⌫. By comparing the expres-sion共20兲 for the coefficient A with Eq. 共16兲 for the transfer

rate␣, we see thatF_⬁= 1 −⌫_␾/ 2␣+O共⌫_␾2兲 if one of the three rates⌫,g,T is much larger than the other two. In this regime, the sensitivity to decoherence is determined entirely by how fast the state can be transferred.

As found in Ref. 7, in connection with the entanglement production, the effect of decoherence on the charge qubit is minimal if T⬇⌫⬇g. More precisely, the fidelity F_⬁is maxi-mized for fixed g and ⌫_␾ if T / g =2₅

冑

5⬇0.89, ⌫/g=2₅

冑

39 ⬇2.50 if ⌫␾Ⰶg and if T/g=12

冑

3⬇0.87 if ⌫␾Ⰷg. As shown

in Fig. 3, the fidelity depends nonmonotonically on ⌫_␾, approaching the asymptotic limit 1₂ from below for T / g ⬎

冑

3 / 2⬇1.22. When F_⬁⬍1₂, the fidelity of the quantum state transfer can be improved by changing the phase of兩−典 by ␲, so that F_⬁哫1−F_⬁. With this procedure the fidelity may actually increase with increasing⌫_␾.

In conclusion, we have analyzed a mechanism for the quantum state transfer from charge qubits to photon qubits, which is deterministic 共no postselection is required兲, and which depends only algebraically on the decoherence rate. The mechanism relies on the coupled dynamics of an electron and a photon in a microwave cavity, but the transfer can be sufficiently fast so that only a few optical cycles of emission-absorption are required. Decoherence rates as large as 10% of the emission rate then do not de-grade the fidelity of the quantum state transfer below about 0.9. These characteristics suggest that the mechanism consid-ered might have promising applications in quantum informa-tion processing.

This research was supported by the Dutch Science Foun-dation NWO/FOM.

1_{J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Rev.}

Lett. 78, 3221共1997兲.

2_{T. M. Stace, C. H. W. Barnes, and G. J. Milburn, Phys. Rev. Lett.}

93, 126804共2004兲.

3_{V. Cerletti, O. Gywat, and D. Loss, Phys. Rev. B 72, 115316}

共2005兲.

4_{M. Titov, B. Trauzettel, B. Michaelis, and C. W. J. Beenakker,}

New J. Phys. 7, 186共2005兲.

5_{T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and Y.}

Hirayama, Phys. Rev. Lett. 91, 226804共2003兲.

6_{J. Gorman, D. G. Hasko, and D. A. Williams, Phys. Rev. Lett. 95,}

090502共2005兲.

7_{C. Emary, B. Trauzettel, and C. W. J. Beenakker, Phys. Rev. Lett.}

95, 127401共2005兲.

8_{Yu. V. Nazarov, Physica B 189, 57共1993兲.} 9_{S. A. Gurvitz, Phys. Rev. B 57, 6602共1998兲.}

10_{B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756共1977兲.} 11_{S. Vorojtsov, E. R. Mucciolo, and H. U. Baranger, Phys. Rev. B}

71, 205322共2005兲.

12_{Z.-J. Wu, K.-D. Zhu, X.-Z. Yuan, Y.-W. Jiang, and H. Zheng,}

Phys. Rev. B 71, 205323共2005兲.

13_{T. Itakura and Y. Tokura, Phys. Rev. B 67, 195320共2003兲.} 14_{S. A. Gurvitz, Phys. Rev. B 56, 15215共1997兲.}

FIG. 3. Decay of the long-time fidelityF_⬁of the quantum state transfer with increasing decoherence rate. We have taken ␣=␤ =1₂

冑

2. The solid curve is for paramter values, at which the quantum state transfer is least sensitive to decoherence.

DETERMINISTIC QUANTUM STATE TRANSFER FROM¼ PHYSICAL REVIEW B 73, 121307共R兲 共2006兲

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