PHYSICAL REVIEW A 80, 033803 ͑2009͒
Qubit coherent control and entanglement with squeezed light fields
Ephraim Shahmoon,1 Shimon Levit,1 and Roee Ozeri2 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel ͑Received 19 January 2009; published 2 September 2009͒ We study the use of squeezed light for qubit coherent control and compare it with the coherent-state control field case. We calculate the entanglement between a short pulse of resonant squeezed light and a two-level atom in free space and the resulting operation error. We find that the squeezing phase, the phase of the light field, and the atomic superposition phase all determine whether atom-pulse mode entanglement and the gate error are enhanced or suppressed. When averaged over all possible qubit initial states, the gate error would not decrease by a practically useful amount and would in fact increase in most cases. However, the enhanced entanglement may be of use in quantum communication schemes. We discuss the possibility of measuring the increased gate error as a signature of the enhancement of entanglement by squeezing.
DOI: 10.1103/PhysRevA.80.033803 PACS number͑s͒: 42.50.Ct, 03.67.Bg, 03.65.Yz, 32.80.Qk
I. INTRODUCTION to, in principle, improve the fundamental limitation in phase ͓ ͔ The coherent control of a two-level atom in free space metrology accuracy 9,10 . The use of squeezed light states was also shown to be beneficial for continuous variable with resonant laser fields is important from both a fundamen- ͓ ͔ tal as well as a practical standpoint. In particular, for quan- quantum key distribution 11–13 . Here, we explore the pos- tum information processing purposes, where the two-level sible advantage of the use of squeezed light for qubit coher- atom encodes a single-quantum bit ͑qubit͒, coherent control ent control. Both the error of a single-qubit quantum gate and the entanglement between the atomic qubit and the SL mode accuracy plays a crucial role. Fault-tolerant quantum error are calculated. correction schemes, necessary for large scale quantum com- Atom-SL interactions were studied extensively in the past puting, demand that the error in a single-quantum gate is ͓ ͔ ͓ ͔ 14,15 . In most cases the atomic dynamics was found to be below a certain, typically very small, threshold 1 . modified in a squeezing-phase-dependent way. Gardiner ͓16͔ In recent years the fundamental limitation to qubit control found that a two-level atom damped by a squeezed vacuum accuracy imposed by the quantum nature of the control light reservoir exhibits squeezing-phase-dependent suppression or ͓ ͔ fields has been the focus of several studies 2–7 . For a co- enhancement of atomic coherence decay. Carmichael et al. herent state of light ͑CL͒, where both phase and amplitude ͓17͔ added a coherent field that drives the atom and showed fluctuations are at the standard quantum limit ͑SQL͒,itwas that the dynamics of the atomic coherence depends on the found that the gate error is determined by the qubit sponta- squeezed vacuum and the coherent driving field relative neous decay rate. At a time shorter than the qubit decay time, phase. Both assumed a broadband squeezed vacuum reser- part of that error can be interpreted as due to entanglement voir whereas the finite bandwidth case was studied in ͓18͔. between the control field mode and the qubit ͑forward scat- Milburn, on the other hand, studied the interaction of a tering͒. single-mode SL with an atom using the Jaynes-Cummings This entanglement is a measure of the information on the model and found a squeezing-phase-dependent increase or atomic state carried away by the pulse field and could, in decrease in the collapse time ͓19͔. principle, be exploited as a resource in quantum communi- Here we assume a qubit that is encoded in an atomic cation schemes. For example, the scheme proposed in ͓8͔ two-level superposition, and a control field that is realized uses the entanglement between an atomic cloud and a pulse with a pulse of SL in a single electromagnetic ͑EM͒ field mode at each node of a quantum network in order to entangle mode and of duration shorter than the typical atomic decay two distant nodes. For a single-qubit coherent control with a time in free space. We find that the squeezing phase, the CL pulse of duration , the entanglement, measured by the phase of the light field, and the atomic superposition phase tangle T, was found to be ͓6͔ Tϳϰ1/ͱ¯n, where ¯n is the all determine whether atom-pulse mode entanglement and average number of photons in the pulse. Thus, in agreement the gate error are enhanced or suppressed. It is found that, with the correspondence principle, quantum entanglement although reduced for certain qubit initial states, when aver- becomes very small as the pulse approaches the classical aged over all possible states the minimal gate error is com- limit, i.e., ¯nӷ1. parable to that with CL fields. It is, therefore, natural to ask whether a control with non- The paper is outlined as follows. Section II describes a coherent light field states would reduce or increase both the general model of the interaction between a light pulse and an operation error and atom-light mode entanglement. In this atom in free space within the paraxial approximation. In Sec. paper we investigate the fundamental quantum limitation to III the pulse mode is assumed to be in a squeezed state and the fidelity of qubit coherent control with squeezed light ͑SL͒ its effect on the atom is examined. Atom-pulse entanglement fields. As opposed to CL, SL has phase-dependent quantum as well as the operation error is calculated. Some aspects of fluctuations, where one field quadrature has reduced, and the possible experimental realization and the relation to quantum other increased, fluctuations compared with the SQL. Re- communication schemes are discussed in Sec. IV, where our duced phase fluctuations of squeezed light have been shown results are also briefly summarized.
1050-2947/2009/80͑3͒/033803͑11͒ 033803-1 ©2009 The American Physical Society SHAHMOON, LEVIT, AND OZERI PHYSICAL REVIEW A 80, 033803 ͑2009͒
II. GENERAL MODEL AND APPROACH general time dependent. The f0 mode coupling is We describe the interaction of a two-level atom with a 0 quantized pulse of light propagating in free space. Classi- g ͑t͒ = dͱ eik0za ͑z − ct͒, ͑3͒ f0 ⑀ ប f0 a cally the propagation of such a pulse is well approximated by 2 0 A ͑ ͒ a paraxial wave equation. A paraxial formalism for quantized where za is the atom’s position. In Eq. 2 we assume both ͓ ͔ ͑ ͒ pulses was described in 4,20,21 . We briefly describe the the envelope of the pulse f z and the dipole matrix ele- paraxial model following a similar approach to that in ͓4͔. 0 ment d to be real, and set za =0. A similar model Hamiltonian The derivation of this approximation is outlined in the Ap- was also used in ͓2,6͔ where the entanglement between a CL pendix, where we start from the basic atom-field Hamil- pulse and an atom was calculated. Note that the time depen- tonian in free space and get to the model described below. dence of the atom-pulse coupling g ͑t͒ is identical to that of f0 A. Pulse mode the pulse; the pulse, therefore, switches the interaction on and off. Consider a laser pulse traveling in the z direction and The relevant bipartite system is composed of the atom and ជ interacting with a well-localized atom or ion at position ra. ˆ the f0 light mode, coupled by HS. The rest of the EM modes This pulse is a wave packet of Fourier modes around some ͕ ͖ ជ except for f0, r , make up a reservoir that interacts with the wave vector k0 =k0zជ, where zជ is a unit vector pointing to the ជ bipartite system via the coupling with the atom. z direction, i.e., it is composed of a carrier wave vector k0 and a slowly varying envelope. The pulse is assumed to propagate with a small diffraction angle so that the paraxial III. SQUEEZED LIGHT PULSE OPERATING ON AN approximation is valid. A natural description for such a pulse ATOM is as a superposition of Gaussian beams with a well-defined We consider a pulse of squeezed light operating on an transverse area A at the position of the atom, each with wave atom. The pulse’s average field changes the atomic superpo- Ӷ number k0 +k, where k k0, analogous to a narrowband wave sition coherently. In a Bloch sphere representation this cor- packet of Fourier modes in one dimension. In this approxi- responds to a rotation of the Bloch vector. The quantum fluc- mation the pulse is effectively a one-dimensional beam with tuations of the field lead to atom-pulse entanglement and add transverse area A similar to a Gaussian beam. The envelope a random component to the average operation. We wish to of the pulse mode, ͑z͒, is a function of width c, where f0 analyze and calculate the error caused by quantum fluctua- is the pulse duration and c is the speed of light. The field tions as well as the degree of atom-pulse mode entangle- operator assigned to this pulse mode is aˆ . f0 ment. The model outlined in Sec. II is sufficiently general to describe a pulse in any quantum state and its interaction with B. Interaction picture and the atom-pulse bipartite system the atom. In ͓2,6͔ calculations for CL state were performed. We start with the full atom-field Hamiltonian in free In this section we will discuss quantum pulses of SL inter- space, acting with a two-level atom. The general formalism is pre- ˆ ˆ ˆ ˆ sented in Sec. III A. In Sec. III B we consider a rotation of H = HF + HA + HAF, the Bloch vector by 180° about an axis lying in the Bloch 1 † ˆ ˆ ប ប ជ sphere equatorial plane ͑a pulse͒ as an example and ne- HA + HF = a ˆ z + ͚ kaˆ kជaˆ k, 2 kជ glect the reservoir ͑i.e., the rest of the EM modes͒. This allows obtaining of analytic expressions for the atom-pulse ˆ ប͑ ͒͑ ͒ ͑ ͒ HAF = ͚ i gkជaˆ kជ − H.c. ˆ + + ˆ − , 1 entanglement, which, in this case, is the sole generator of ជ k gate error. For longer pulses, reservoir effects become in- where kជ are the indices of a Fourier mode and its polariza- creasingly dominant for both entanglement and error genera- tion. In Sec. III C we obtain the results in the presence of the tion, respectively, and gkជ are the atom-mode couplings. ប reservoir, i.e., including spontaneous emission into all Here the ˆ operators are the spin-1/2 Pauli matrices and a is the energy separation between the two atomic levels. Mov- modes. ing to the interaction picture with respect to the interaction ˆ A. Unitary transformations and master equation HAF in the rotating wave approximation, assuming no detun- ing 0 =k0c= a, and using the paraxial approximation, we In order to determine the final state of the bipartite photon get the Hamiltonian in the interaction picture ͑see Appendix͒ pulse-atom system in the presence of the reservoir of free ˆ ˆ ˆ EM modes, an appropriate master equation should be used. HI = HS + HSR, 1. Initial state ˆ ប ͑ ͒͑ † ͒ HS = i gf t aˆ f ˆ + − aˆ f ˆ − , 0 0 0 Here we assume the light mode f0 to be a single pulse mode in a squeezed state and we choose the following initial ͒ ͑ ͒ † ء ប ͑ ˆ HSR = i ͚ graˆ r ˆ + − gr aˆ r ˆ − , 2 state for the EM field r ͉␣ ͘ ͉ ͘ ͑ ͒ , f 0 ͕r͖. 4 where the indices ͕r͖ denote all the free space EM modes 0 apart from the f0 pulse mode, with couplings gr that are in Here
033803-2 QUBIT COHERENT CONTROL AND ENTANGLEMENT WITH… PHYSICAL REVIEW A 80, 033803 ͑2009͒
͑ Ͻ Ͻ ͒ϵ ϵ ͱ/ ͉␣,͘ = Dˆ ͑␣͒Sˆ͉͑͒0͘ , ͑5͒ gf 0 t g , f0 f0 0 where ͑ Ͻ Ͼ ͒ ͑ ͒ gf t 0,t =0. 10 †2͒ 0 2 ء͑͒ / ͑ ˆ͑͒ϵˆ ͑͒ 1 2 aˆ f − aˆ f ͑ ͒ S Sf = e 0 0 6 0 2 /͑ ប ͒ Here =d 0 2 0 Ac is the atomic decay rate into all the is the squeezing operator with the squeezing parameter paraxial modes of the same transverse profile as the f pulse 0 =rei2 and mode ͓4͔. The dynamics during the pulse is then approxi- mately governed by ء␣ † ␣ ˆ ˆ aˆ − aˆ f D͑␣͒ϵD ͑␣͒ = e f0 0 ͑7͒ f0 ˆ ˆ ˆ is the displacement operator with the field average ␣=͉␣͉ei . H = HS + HSR, The phases and together with the phases associated with the initial state of the atomic qubit will play an impor- tant role in the following. Hˆ = iបͱ cWˆ + iប ͑eiˆ − e−iˆ ͒, S 2 + − 2. Unitary transformations ͒ ͑ ͒ † ء ប ͑ ˆ It is convenient to transform the problem into a represen- HSR = i ͚ graˆ r ˆ + − gr aˆ r ˆ − , 11 tation where the initial state of the pulse mode is the vacuum r state. To achieve this, we first use the Mollow transformation ˆ ͓22,23͔ and move the average field ␣ from the initial state where W is the dimensionless operator that describes the ˆ quantum field part of the atom-pulse interaction, into the Hamiltonian. This affects only the term HS in Hamil- tonian ͑2͒, which becomes s s ˆ ͫ ͩ −2i †ͪ ͩ † 2i ͪͬ ͑ ͒ W = ˆ + aˆ − e aˆ − ˆ − aˆ − e aˆ . 12 Hˆ = iបg ͑t͓͒aˆ ˆ − H.c.͔ + iបg ͑t͓͒␣ˆ − H.c.͔. ͑8͒ c c S f0 f0 + f0 + We then make a time-independent Bogoliubov transforma- The corresponding master equation for the bipartite system ˆ †͑͒ ˆ ˆ͑͒ ˆ ˆ͑ ͒ tion, S HS . This again affects only HS, which becomes density matrix, t , in the presence of the reservoir is
Hˆ = iបg ͑t͓͒ˆ ͑caˆ − e−2isaˆ †͒ − H.c.͔ + iបg ͑t͓͒␣ˆ dˆ i ⌫ S f0 + f0 + =− ͓Hˆ ,ˆ͔ − ͕ˆ ˆ ,ˆ͖ + ⌫ˆ ˆˆ , ͑13͒ ប S + − − + ͔ ͑ ͒ dt 2ء␣ − ˆ − . 9 Here aˆ ϵaˆ , s=sinh͑r͒, and c=cosh͑r͒. In this representa- where ⌫ is the atomic decay rate in free space. f0 tion the initial state of all the EM modes is the vacuum state. This master equation assumes a Markovian reservoir, i.e., Hamiltonian ͑9͒ includes two types of driving terms. The a reservoir with a very short correlation time. Such assump- second term in Eq. ͑9͒ describes an atom driven by the av- tion is not obvious for the free space EM reservoir that lacks erage field and, therefore, contains no EM field operators. a single pulse mode, such as the one we consider. This rather The first term in Eq. ͑9͒ describes interaction with the EM artificial reservoir has a correlation time of the order of the ͓ ͔ squeezed vacuum. Unlike the coherent-state case, it includes pulse duration . In Sec. IVB of 6 , a similar Markovian terms such as aˆ †ˆ similar to those that are typically ne- equation was written for a CL pulse and was then solved + Ӷ ͓ ͔ glected in the rotating wave approximation. Here, however, perturbatively for 1. In fact, in 24 it was found that the ͑ ͒2 Ӷ such terms appear as a result of squeezing, and indeed vanish assumption g = 1 is actually a necessary condition for r=0. These terms reflect the fact that unlike the ordinary for the validity of the Markovian assumption and master ͑ ͒ Ӷ⌫ vacuum the squeezed vacuum contains photons that can be equation 13 , together with the condition . An intuitive ͑ ͒2 Ӷ absorbed by the atom. argument for the requirement g = 1 can be based on the time scales related to the pulse mode that is excluded from the reservoir. While the correlation time of the reservoir 3. Master equation is , this correlation’s effect on the dynamics should be of the We now examine a particular case of a pulse of dura- order of . Therefore, Markovian dynamics is obtained by tion . In a Bloch sphere representation and within our nota- requiring Ӷ1. Markovian time evolution guarantees that tions, the average field of such a pulse rotates the Bloch the probability of successive reabsorptions and reemissions vector by 180° about an axis, lying in the sphere equatorial of photons from and into the f0 mode, typical of the Jaynes- plane, in an angle from the y axis. For a constant interac- Cummings dynamics, are negligible as it should be in free tion coefficient g this requires ͉␣͉g=/2. The time- space. dependent interaction coefficient g ͑t͒ from Eq. ͑3͒ includes ˆ͑ ͒ ͉ ͗͘ ͉ ͉͘ f0 The initial bipartite state, 0 = ,0 ,0 , where de- the pulse profile, which is assumed to be approximately con- notes any atomic qubit state and f0 is in the vacuum 0, is stant during the time interval and vanishing elsewhere. We pure and separable. Solving Eq. ͑13͒ for t= yields the bi- set t=0 to be the time at which the pulse reaches the atom so partite density operator ˆ͑͒ from which properties such as at t= it has already passed it, atom-pulse entanglement can be calculated.
033803-3 SHAHMOON, LEVIT, AND OZERI PHYSICAL REVIEW A 80, 033803 ͑2009͒
B. Entanglement and error probability calculations neglecting pand the normalization factor n͑͒=1/ͱ͉͗͘ to second or- the reservoir der in and use it to normalize the state. We start by considering a pulse duration that is much 2. Entanglement calculation shorter than the atomic decay time ⌫−1. An analytic solution to Eq. ͑13͒ can be found in this case by neglecting the res- We measure the entanglement of the bipartite state using ervoir. In this short pulse limit, the probability to absorb a the tangle monotone, which is related to the square of the ͓ ͔ photon from the pulse and then to emit one to the reservoir is concurrence 25,26 , and compare our results for SL pulses ͓ ͔ negligible. Then, one can indeed neglect any modification to with those obtained in 6 for CL pulses. ͉͘ the atom-pulse entanglement due to the presence of the res- For a pure bipartite state of two subsystems A and B, ervoir. However, gate error calculated in this approximation and where one of the subsystems is a two-level system, the ͓ ͔ is induced only by the quantum fluctuations of the pulse tangle takes the form 27 mode. We note that, at least for the CL case, this error is a ͉͉͑͗͒͘ ͑ ͕͓ ͉͉͔͑͗͒͘2͖͒ ͑ ͒ T =2 1−TrA TrB . 17 relatively small part of the total error, which is mostly due to spontaneous photon emission to the reservoir EM modes. We calculate the atom-pulse tangle for various atomic initial states up to O͑2͒, by inserting the normalized final state, 1. Bipartite wave-function calculation calculated using Eq. ͑16͒, into Eq. ͑17͒. ͉ ͘ ͉ ͘ Neglecting the reservoir, Eq. ͑13͒ becomes a Schrödinger For atomic initial states g and e we get equation, the solution of which at t= is ͉͑͒͘=Uˆ ͉͑͒,0͘, s 2 s T Ϸ ͫ1+ͩ ͪ −2ͩ ͪcos͑2͒ͬc2. ͑18͒ where c c ͑ /ប͒ ˆ ͑/ ͒͑ ͒ ˆ Uˆ ͑͒ = e− i HS = e 2 ˆ +− ˆ − + W ͑14͒ For a CL pulse, i.e., s=0 and c=1, the tangle is given by T Ϸ, which is the same as calculated in ͓6͔. For nonzero is the propagator for t= . Here =0 is chosen and squeezing this result is rather strongly modified and in par- ϵ ͑ ͒ͱ Ӷ cosh r . We now take the assumption 1 one step ticular the tangle depends on the squeezing phase .Asan Ӷ further and require 1. This puts a limit on the magnitude example when er ӷ1 and, therefore, s/c is close to unity, we of the squeezing parameter r. As will be shown later, in the obtain TϷ2c2͓1−cos͑2͔͒. This expression has a typical paraxial approximation, where is small, and with presently interference pattern, which will be discussed in Sec. IIIB4. available squeezing technology, this is a very good approxi- Let us now consider two different squeezing phases, ͑ ͒ mation. Looking at the propagator in Eq. 14 one notes that =0 and =/2, which for =0 correspond to amplitude and initially the matrix elements of Wˆ are O͑1͒ so Uˆ ͑͒ can be phase squeezing, respectively. We obtain from Eq. ͑18͒ expanded in powers of T Ϸ eϯ2r, ͑19͒ ˆ ͑͒ ˆ ͑0͒ ˆ ͑1͒ 2 ˆ ͑2͒ ͑ ͒ U = U + U + U + ¯ , 15 where the minus ͑plus͒ sign is for =0 ͑=/2͒. When where compared with the CL case, in the =/2 case we get en- hancement whereas in the =0 case we get suppression of ˆ ͑0͒ ˆ ͑ ͒ U = U0 =1 , entanglement. This result is plotted in Fig. 1, where we set =0.1⌫−1 and =⌫/1000. This estimate for results from 1 assuming a Gaussian beam focused on the atom with a lens ˆ ͑1͒ ͵ ˆ ͑ ͒ ˆ ˆ ͑͒ U = d U0 1− WU0 , of numerical aperture of about 0.l, consistent with the 0 paraxial approximation ͓2,6͔. With these numbers, the as- sumption Ӷ1 is valid for rՅ3. Current state-of-the-art 1 1 light squeezing experiments do not exceed r=1.16 ͑see Sec. ˆ ͑2͒ ͵ ͵ ˆ ͑ ͒ ˆ ˆ ͑ ͒ ˆ ˆ ͑ ͒ U = d 2 d 1U0 1− 1 WU0 1 − 2 WU0 2 , IV͒; therefore, Ӷ1 is a very good assumption. 0 2 We repeat the tangle calculation for initial superposition 1 i states of the form ͱ ͑e A͉g͘+͉e͒͘. In a Bloch sphere repre- ͑ ͒ 2 ..., 16 sentation this state is represented by a vector lying in the ˆ ͑͒ϵ ͓͑/ ͒͑ ͔͒ equatorial xy plane with an angle A from the positive x and where we defined U0 exp 2 ˆ + − ˆ − . Using this approach we calculate the final states for vari- direction. The tangle we get is 2 ous initial states of the atom. We find that up to O͑ ͒ it is −i2 A e i2 ϯr i Ϯr 2 enough to work in the truncated bipartite basis that includes T Ϸ ͓͑1+e A͒e +4e Ae ͔ , ͑20͒ 42 only four states: ͕͉g,0͘,͉g,1͘,͉e,0͘,͉e,1͖͘. States of the form ͉,2͘ do appear in the expansion. However, in our cal- where the upper ͑lower͒ sign refers to =0 ͑=/2͒. This culations of the entanglement and gate error they do not expression is plotted in Fig. 1 for the two initial states A ͑2͒ / contribute up to and including O , and are, therefore, =0 and A = 2. For each initial state, a squeezing-phase- omitted. We thus end up with a truncated basis that, follow- dependent enhancement and suppression compared with the ing the perturbative assumption, includes only a single pho- CL case are observed. tonic excitation in the pulse mode, as argued in ͓6͔. The As could be expected from atom-SL interaction, we ob- resulting final state ͉͘ is not normalized; we, therefore, ex- serve dependence on the relative phase between the squeez-
033803-4 QUBIT COHERENT CONTROL AND ENTANGLEMENT WITH… PHYSICAL REVIEW A 80, 033803 ͑2009͒
−1 −2 10 10 φ=π/2 φ=0 −2 10
−3 −3 10 10 |g〉 or |e〉, φ=0
|g〉+|e〉, φ=0 > −4 i|g〉+|e〉, φ=0 10 error angle |g〉+|e〉, φ=π/2 P T < i|g〉+|e〉, φ=π/2 −5 −4 −4.62 10 |g〉 or |e〉, φ=π/2 10 10
−6 10 −4.67 10 0 0.5 −7 −5 10 10 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 squeezing (r) squeezing (r)
FIG. 1. ͑Color online͒ The tangle ͓Eq. ͑17͔͒ calculated for the FIG. 2. ͑Color online͒ The average error probability of an op- atom and the squeezed light pulse mode bipartite system vs the eration with a squeezed light pulse on an atom vs squeezing squeezing strength. The pulse duration is =0.1⌫−1 and the cou- strength. The average is over all possible atomic initial states. As in −1 pling to the rest of the EM modes is neglected. The decay rate into Fig. 1, =0.1⌫ and =⌫/1000 are taken. For most cases, the paraxial modes is taken to be =⌫/1000. Results are shown for average error with squeezed light increases relative to coherent light four atomic initial states and for the squeezing phases =0,/2, ͑r=0͒. For the squeezing phase =0 ͑amplitude squeezing͒ and corresponding to amplitude and phase squeezing, respectively. squeezing strength 0ϽrϽ0.45 a very small decrease in the average Compared with the coherent light case ͑r=0͒, one obtains a error is observed, as shown in the figure inset. squeezing-phase-dependent suppression and enhancement of the en- tanglement as squeezing gets larger. For coherent control in cases where the qubit initial state is a priori known, the squeezing phase can be chosen such ing phase, , and the phase of the average of the pulse, that it will decrease the operation error. Examples for such a ͑which is chosen here to be zero͒. Furthermore, the tangle case include qubit initialization to states other than those indeed depends on the atomic initial state. For fixed pulse reachable by optical pumping techniques or the pulses in phase, the change in entanglement for initial states that lie in Ramsey spectroscopy of separated fields. the Bloch sphere equatorial xy plane depends on the differ- For quantum information processing purposes the relevant ence between the squeezing phase and A. error is that averaged over all possible atomic initial states, ͗ ͘ To conclude, the results imply that for any given qubit Perror . The averaged error is calculated analytically using initial state the squeezing phase can be appropriately chosen the fidelity of six different initial atomic states ͓28͔ to either enhance or suppress the qubit-pulse entanglement. ͗ ͘Ϸ͑ Ϯ2r ϯ2r͒ ͑ ͒ Perror 0.0675e + 0.1665e , 22 3. Error probability calculation where as before the upper ͑lower͒ sign is for =0 ͑ / ͒ The error probability of a quantum operation is just one = 2 . This result is plotted as a function of squeezing in ͗ ͘ minus the probability of the atomic qubit to be in the target Fig. 2. As can be seen the averaged error Perror depends on state ͉͘ of the operation after the operation is performed. A the squeezing phase . For most values of the squeezing classical light pulse can be chosen to transform an atom from parameter r the average error is larger than in the CL case ͑ ͒ the initial to a prescribed final state ͉͘. When operating with r=0 . This implies that on average, more entanglement is the quantized pulse, the operation is subject to quantum fluc- generated between the atom and the pulse in SL than in CL Ͻ Ͻ tuations. The fidelity F is the probability of the atom to be in case. Still one observes that for the region 0 r 0.45, and ͉͘ ͉͗ ͉͘ =0 ͑amplitude squeezing͒, the average error drops relative after the quantized field has operated on it, F= ˆ A , ͑ to the error in the CL case. Though practically small, in this where ˆ A is the density matrix that describes the generally mixed͒ final state of the two-level atomic system. The error range of the squeezing parameter the SL operation has lower probability is then, quantum limit to the operation error than with the CL pulse due to reduced intensity noise. P =1−F =1−͉͗ˆ ͉͘. ͑21͒ error A 4. Discussion of the results Since in the absence of the reservoir the sole error is that, ˆ due to atom-pulse entanglement, we were able to find a Quantum interference. Consider the interaction W be- tween the atom and the fluctuating part of the light pulse simple analytic relation between the tangle and the error ͑ ͒ ͑2͒ Ϸ shown in Eq. 12 . In addition to the Jaynes-Cummings-like probability calculated up to O , T 4Perror. This linear Wˆ ϵi͑aˆˆ ͒ relation implies that the results for Perror follow similar atom-photon interaction term 1 + −H.c. , present also ˆ trends to those shown for the tangle in Fig. 1, and are squeez- in the CL case, it contains a squeezing-dependent term W2 ϵ ͓͑ / ͒ −2 † ͔ ing phase and initial atomic state dependent. i s c e ˆ +aˆ −H.c. . Atomic and light mode excitations
033803-5 SHAHMOON, LEVIT, AND OZERI PHYSICAL REVIEW A 80, 033803 ͑2009͒ can be generated via both Wˆ and Wˆ . These two paths in- 1 1 2 ␣ ϵ ͑x + ix ͒. ͑24͒ terfere quantum mechanically resulting in a dependence of 2 1 2 atom-light entanglement and error probability on both ␦ squeezing and atomic phases. Here X1,2 are Hermitian operators with zero average and Starting from ͉g,0͘ we examine the bipartite state evolu- x1,2 are real numbers representing the averages of the quadra- tion under the first two operators in Eq. ͑16͒. The classical tures. We then define the vectors ˆ ͑0͒ ͉ ͘ field term, U , is evolving the state to e,0 via superposi- xជ ϵ͑x ,x ,0͒, tions of ͉g,0͘ and ͉e,0͘. Simultaneously the state evolves 2 1 ˆ ˆ ͑1͒ ˆ under the term W in U . The W1 part of it couples only to ជ␦ ϵ͑␦ ␦ ͒ ͑ ͒ ͉ ͘ ͉ ͘ ˆ X X2, X1,0 , 25 e,0 evolving it to g,1 . The W2 part on the other hand couples only to ͉g,0͘ and transforms it to ͉e,1͘. Subsequent so that Hamiltonian ͑8͒ becomes evolution under Uˆ forms superpositions of ͉e,1͘ and ͉g,1͘, 0 g the amplitudes of which are coherent sums ͑and, therefore, Hˆ =−ប ͑xជ + ជ␦X͒ · ជˆ , ͑26͒ ͒ ˆ 2 show interference of evolution amplitudes under both W1 ˆ ˆ ͑ ͒ and W2. Quantitatively, with ជ = ˆ x , ˆ y , ˆ z . The resulting Heisenberg equation for ជˆ ͑t͒ is ͑ ͒ ͑ ͒ ͉͑͒͘Ϸ͓Uˆ 0 + Uˆ 1 ͔͉g,0͘ dជˆ g g =− xជ ϫ ជˆ − ជ␦X ϫ ជˆ , ͑27͒ 1 dt 2 2 ͉ ͘ ͵ ˆ ͑ ͒͑ ˆ ˆ ͒ ˆ ͉͑͒ ͘ = e,0 + d U0 1− W1 + W2 U0 g,0 0 with all the operators written in the Heisenberg picture. ¯ / ͑ ͒ 1 Rescaling the time to t=t Eq. 27 becomes = ͉e,0͘ − ͵ dUˆ ͑1−͒sinͩ ͉ͪg,1͘ 0 2 dជˆ 1 0 =−⍀ជ ϫ ជˆ − ជ␦X ϫ ជˆ , ͑28͒ ¯ 2 1 s dt ˆ ͑ ͒ −i2 ͩ ͉ͪ ͘ − ͵ d U0 1− e cos e,1 ជ 0 c 2 where ⍀=͑gxជ /2͒ is the dimensionless Rabi vector and =g. Recall that for a pulse ͉␣͉=͑/2g͒ and, therefore, 1 s 1 s ជ = ͉e,0͘ − ͩe−i2 +1͉ͪe,1͘ + ͩe−i2 −1͉ͪg,1͘, ͉⍀͉=O͑1͒. Our calculation also assumes =cosh͑r͒Ӷ1, c 2 c i.e., even for matrix elements of ជ␦X, which are of order ͑23͒ cosh͑r͒, ជ␦X is still very small. We can then expand ជˆ ͑¯͒ ជˆ ͑¯͒ ជˆ ͑¯͒ ជ␦ ͑¯͒ ជ␦ ͑¯͒ the operators t = 0 t + 1 t +¯ and X t = X0 t where Uˆ ͉͑͒g/e,0͘=cos͑ ͉͒g/e,0͘Ϯsin͑ ͉͒e/g,0͘. 0 2 2 ជ␦ ͑¯͒ ˆ ˆ + X1 t +¯, to obtain to first order in , The interference between evolution under W1 and W2 is ͉ ͘ ͉ ͘ clearly seen in the amplitudes of e,1 and g,1 . For large dជˆ 1 2 Ϸ ⍀ជ ϫ ˆ ជ␦ ϫ ˆ ͑ ͒ squeezing, i.e., sХc, and =0,weget͉e͘ ͉͑0͘− ͉1͒͘, − ជ − X0 ជ 0. 29 ¯ 2 which contains no entanglement. For =/2 we get ͉e,0͘ dt +͉g,1͘, which is entangled with entanglement strength on ͱ The last term includes the cross product of the Heisenberg the order of the interaction strength =cosh͑r͒ . The en- picture operators in the absence of atom-field interaction, so tanglement, therefore, increases exponentially with squeez- that ជˆ ͑¯t͒ is simply the solution of the OBE with the classical ing, which might prove useful for quantum repeaters as will 0 field ⍀ជ and describes evolution in the atomic part of the be discussed in Sec. IV. This result is similar to that obtained ជ␦ ជ␦ ͑ ͒ for the tangle in Sec. IIIB2. For other atomic initial states Hilbert space, and X0 = X ¯t=0 is in the photonic part of similar calculations up to second order in were performed the Hilbert space. leading to the results shown in Secs. IIIB2and IIIB3. The average over atomic degree of freedom can be de- ˆ ͗ ˆ ͘ ϵ ͓ ˆ ͔ Optical Bloch equation (OBE) with a noisy field. An in- fined on any operator O to be O A trA AO , where A de- tuitive understanding of the error results from Sec. III B can notes the atom system and A is the reduced atomic density be obtained from the Bloch sphere picture. To this end we operator. Averaging Eq. ͑29͒ this way we get an OBE with a derive the OBE for a one mode SL field, starting from noisy field, ͔ ͒ء␣ † ͑ ប ͓͑ ␣͒ ˆ ͒ ͑ Hamiltonian 8 , H=i g aˆ + ˆ + − aˆ + ˆ − . As usual d͗ជˆ ͘ 1 ˆ ␦ A Ϸ ⍀ជ ϫ ͗ˆ ͘ ជ␦ ϫ ͗ˆ ͘ ͑ ͒ for SL, we define the quadrature operators X1,2= X1,2+x1,2 − ជ A − X0 ជ 0 A. 30 with dt¯ 2 ͗ˆ ͘ ͑ ͒ ជ␦ Note that ជ A ¯t and X0 are operators in the photonic space 1 ͗ជˆ ͘ ͑¯͒ aˆ ϵ ͑␦X + i␦X ͒, and are, therefore, fluctuating, whereas 0 A t is just the 2 1 2 c-number Bloch vector solution of an OBE driven by the
033803-6 QUBIT COHERENT CONTROL AND ENTANGLEMENT WITH… PHYSICAL REVIEW A 80, 033803 ͑2009͒
z −2
−3
−4 y −5 T 10 og l −6 x −7 |g〉+|e〉, φ=0 FIG. 3. The driving field in the optical Bloch equation represen- |g〉+|e〉, φ=π/2 ជ tation. The solid arrow is the Rabi vector ⍀ proportional to xជ from −8 i|g〉+|e〉, φ=0 ͑ ͒ ͑ ͒ i|g〉+|e〉, φ=π/2 Eq. 25 for x2 =0 =0 . The error bars are the variances of the ⌬␦ ⌬␦ −9 field fluctuations X2 and X1 in the x and y directions, respec- −4 −3 −2 −1 0 1 ⌬␦X Ͻ⌬␦X log Γτ tively. Here =0 is taken, i.e., 1 2. 10
FIG. 4. ͑Color online͒ The tangle generated between the atom average field ⍀ជ , and it in fact equals to the average of ជˆ ͑¯t͒ and the squeezed light pulse mode as a function of the pulse dura- over both atomic and photonic degrees of freedom. The term ⌫−1 tion scaled to atom free space decay time . The squeezing of the ជ␦ with X0 is the one driven by the fluctuations of the field, pulse mode is taken to be r=1.16, which corresponds to ͑ 2r͒ which in our case are quadrature dependent SL fluctuations. 10 log10 e =10 dB quadrature noise reduction. As in Fig. 1, ⌬␦ =⌫/1000 is taken. The reservoir of the free space EM modes is For =0, the uncertainties of the quadratures are X1 included. Results for two atomic initial states are plotted. The effect −r Յ ⌬␦ r Ն ␦ =e 1 and X2 =e 1, i.e., X1 has reduced fluctuations of the reservoir is seen in the deviation from the initial linear part of ␦ and X2 has increased fluctuations relative to CL. For the plots. This deviation begins as expected at times slightly shorter =/2 we get the opposite. than ϳ⌫−1, when entanglement is enhanced by SL ͑all cases here, We can now present the field in the Bloch sphere as seen apart from i͉g͘+͉e͘, =/2͒. When entanglement is suppressed by in Fig. 3. Taking real ␣ ͑=0͒ and squeezing phase =0, SL, the effect of the reservoir is seen already at much smaller . ͑ ͒ the Rabi vector is pointing to the y direction x2 =0 and is ͑⌬␦ Ͻ ͒ ͑ ͒ ⌫/ amplitude squeezed X1 1 . Now consider an initial Eq. 13 numerically in this basis using = 1000 and cal- ͗ˆ ͘ culate the density matrix ˆ͑͒ of the bipartite final state. We ground state, i.e., ជ A is pointing to the −z direction. For a pulse operation, the xy phase of the field is not relevant, and, then use ˆ͑͒ to evaluate the tangle and the error probability. therefore, phase noise of the field does not affect the opera- tion. The only relevant noise is the amplitude noise, which 1. Tangle calculation for =0 is reduced relative to CL case. We thus expect to get Within the truncated basis, the pulse mode is also a two- a lower error in this operation as compared with the CL level system. We follow ͓25,26͔ where a procedure for the operation. For =/2, the field has an increased amplitude tangle calculation of a two-qubit mixed state is given. Figure ͑⌬␦ Ͼ ͒ noise X1 1 , which results in an increased error com- 4 presents the tangle as a function of the pulse duration .We pared to the CL case. Now consider the y axis pointing initial assume squeezing of r=1.16, which corresponds to ͑ 2r͒ state again for =0. Since the Rabi vector is also pointing to 10 log10 e =10 dB pulse mode quadrature noise reduction. the y direction the average “moment” obtained from the The interaction with the reservoir changes atom-pulse en- ⌫տ cross product is zero. Here only the ␦X “phase” noise, in the tanglement when 1 and should generally decrease it for 2 ⌫ӷ x direction, affects the qubit state. When this noise is in- 1. Whenever the tangle is not suppressed by the SL ⌫−1 creased ͑=0͒ the error is larger than in the CL case, and operation, pulses shorter than 0.1 are well accounted by when it is reduced ͑=/2͒ the error gets lower. For an neglecting the reservoir, i.e., extending the initial linear part initial state pointing to the x direction, one can see that in of the plots. When SL operation suppresses the entangle- ␦ ␦ ment, the interaction with the reservoir becomes more domi- general, both X2 and X1 affect the operation, and, there- nant, and the effect of the reservoir can only be neglected for fore, squeezing in any quadrature typically increases the op- pulses shorter than 0.01⌫−1. eration error. 2. Error probability calculation C. Entanglement and error probability calculations in the The error probability is calculated using Eq. ͑21͒. Figure presence of the reservoir 5 shows the results for =0.1⌫−1 pulse duration. It compares We now wish to examine how accounting for the sponta- the error probability results calculated considering the reser- neous emission into the reservoir of empty modes will voir ͑right-hand part, denoted as IR͒ with the ones calculated modify the above results. To this end the full master equation in Sec. IIIB3where the reservoir was neglected ͑left-hand ͓Eq. ͑13͔͒ has to be solved. We again assume Ӷ1 and work part, denoted as NR͒. Let us now compare between the two 1 ͕͉g ͘ ͉g ͘ ͉e ͘ ͉e ͖͘ ͱ ͑ ͉ ͘ ͉ ͒͘ in the truncated basis ,0 , ,1 , ,0 , ,1 . We solve cases for the initial state 2 i g + e . For r=0, i.e., a CL
033803-7 SHAHMOON, LEVIT, AND OZERI PHYSICAL REVIEW A 80, 033803 ͑2009͒
−2 10 0.029 atom may be used as a quantum communication resource, −3 and, from a more fundamental point of view, is a quantum 10 0.028 −4 phenomenon that would be interesting if observed experi- 10 〉 〉 〉 〉 −5 i|g +|e 0.027 i|g +|e mentally. 10 NR IR −6 10 0.026 The enhancement of entanglement by SL might be of ben- Error Prob. −7 Error Prob. 10 0.025 efit for quantum communication based on quantum repeaters −8 ͓ ͔ 10 0.024 such as the scheme described in 8 . In this scheme atomic 0 1 2 3 0 1 2 3 squeezing (r) squeezing (r) clouds serve as nodes of a quantum network and the com- −1 10 0.076 munication protocol relies on the ability to entangle pairs of distant nodes. This is achieved by first inducing Raman tran- −2 0.074 10 sitions in each cloud, which effectively couple the symmetric 0.072 |g〉+|e〉 −3 collective atomic mode to the forward-scattered Stokes light 10 IR 0.07 mode. The resulting atom-photon entangled state is −4 〉 〉
|g +|e Error Prob. Error Prob. 10 NR 0.068 ͉ ͘ ͉ ͘ ͱ ͉ ͘ ͉ ͘ ͑ ͒ −5 0 a 0 p + P E a 1 p, 31 10 0.066 0 1 2 3 0 1 2 3 squeezing (r) squeezing (r) where E denotes a single excitation of the collective atomic mode, and P is related to the interaction and entanglement FIG. 5. ͑Color online͒ The error probability of an operation with strength between the cloud and the Stokes mode. Then, by a squeezed light pulse on an atom vs squeezing strength. The projecting on the photonic states of the Stokes modes scat- pulse duration is =0.1⌫−1 and atomic decay into all the reservoir tered from two separate clouds, it is possible to nearly maxi- modes is either neglected ͑NR͒ or included ͑IR͒, providing a com- mally entangle the two atomic clouds with probability P. The parison between the two cases. As in Fig. 1, =⌫/1000 is taken. entangled atom-photon state in Eq. ͑31͒ is reminiscent of the The calculation results for two initial state, i͉g͘+͉e͑͘upper hand entangled atom-photon state from Eq. ͑23͒ with =/2, ͒ ͉ ͘ ͉ ͑͘ ͒ part and g + e lower hand part , are shown. The dashed lines are where the atom-photon entanglement P can be increased / for =0 and the solid lines for = 2. with squeezing. This suggests that at least in principle the creation rate of entangled pairs of nodes can be increased by pulse, the error of about 0.0244 seen in the IR case is actu- squeezing the vacuum background of the Stokes mode. Re- ally due to spontaneous emission to all the EM modes, as can call that, however, in order to utilize the atom-photon en- be understood from the Mollow picture ͑see Ref. ͓5͔͒. The tanglement in order to entangle the two nodes, it is needed to corresponding error in the NR case is only 10−5. This sug- project on the photonic states. In the squeezing case, the two ͑ ⌫/ ͒ ͉ / ͘ ͉͑͒ / ͘ gests that, owing to its small solid angle = 1000 , the states 0 1 p are actually S 0 1 so projecting over them error due to coupling to the laser mode is overwhelmed by with a simple photodiode requires first to unsqueeze them. that due to coupling to the reservoir modes. When squeezing An efficient implementation of single-mode squeezing for is introduced ͑rϾ0͒, for =/2, the atom-pulse entangle- quantum repeaters in this context is left for future work. ment is suppressed as seen in the NR case. This suppression Other methods to entangle qubits using broadband squeezed is hardly noticeable when the reservoir modes are included reservoirs were discussed, e.g., in ͓29,30͔. as one can see in the IR plot. Numerically, for r=3,wefind The observation of entanglement between a macroscopic that the error decrease is 0.95ϫ10−5. This matches what mode of light and a single atom is experimentally challeng- could be expected from the NR plot and demonstrates that ing. Contrary to CL, where atom-light entanglement is prob- the reservoir is the major cause of error in this case. On the ably too small to be detected, SL with a large squeezing other hand, for =0, an increase in the error seen in the NR parameter r and a right choice of phases will eventually en- case persists also in the presence of the reservoir, as can be hance SL-atom entanglement to experimentally detectable seen in the IR plot. For very large squeezing this increase values. A direct measurement of entanglement would be may become of the same order of magnitude as the error due through complete state tomography of the combined SL- to the spontaneous emission to all the EM modes. For ex- atom state. The tangle would then be calculated from the ample, for r=3 the increase seen in both NR and IR plots is reconstructed density matrix. The requirements for this ex- ϳ4ϫ10−3, about 16.3% of the 0.0244 error induced by periment would be a squeezed light source with a sufficiently spontaneous emission. large squeezing parameter ͓31͔, atomic state detection, and photodetectors with sufficient efficiency. Alternatively, ignor- IV. DISCUSSION ing the light mode, the error in the final atomic state can be measured for various parameters, such as different squeezing The results of Sec. III show that the entanglement be- phases, thus relaxing the need for an efficient photodetector. tween the pulse mode and the atom is indeed modified in a Examples for squeezed light sources include four wave squeezing-phase-dependent way. For a correct choice of rela- mixing in a nonlinear medium, optical parametric amplifiers, tive phases the error in qubit coherent control can be reduced second-harmonic generation, and amplitude squeezed light with respect to the CL case. However, the use of squeezed from diode lasers among others ͓32͔. All these sources can light would not reduce the average error, relevant for quan- be pulsed, e.g., via electro-optic modulators. The initial light tum computing purpose, by a practically useful amount. On state we assumed for the control field was a single-mode the other hand, the entanglement between SL and a single squeezed state. Parametric down conversion ͑PDC͒ of short
033803-8 QUBIT COHERENT CONTROL AND ENTANGLEMENT WITH… PHYSICAL REVIEW A 80, 033803 ͑2009͒ pulses could be considered as one example of a realization relations. These entanglement enhancement and suppression for such a light source. In ͓33,34͔ the multimode, pulsed SL effects naturally become stronger with squeezing and can get produced by PDC is analyzed and described by a discrete set to a few orders of magnitude. For quantum information pro- of squeezed modes. In fact, for some conditions of phase cessing purposes the relevant error is that averaged over all matching and pump pulse duration, only one of the modes of possible initial states. We have shown that although in prin- the discrete set is squeezed, resulting in a pulse in a single- ciple the average error can be reduced by using SL within a mode squeezed state. specific range of parameters, the resulting minimum error, Even the most advanced single-mode squeezed light however, would be of the same order of magnitude as that sources to date cannot reach quadrature noise suppression ͑ ͓͒ ͔ with CL. In fact, even when not averaged but when taking greater than 10 dB r=1.16 32 . This means that the maxi- into account the free space environment, the error is still mal error increase and tangle expected, and, therefore, the mostly dominated by the atomic decay to the EM reservoir minimal required detection error, will be of the order of a owing to a very small atom-pulse spatial overlap so that any few 10−4 ͑see Figs. 1 and 2͒. Current state-of-the-art atomic reduction is not likely do be detected experimentally. On the state detection has an error that is of similar magnitude ͓35,36͔. However, the efficiency of even the most advanced other hand, the effect of enhanced atom-pulse entanglement photodetectors is far from satisfying such low error rates. It, is of scientific interest and might even prove useful as a therefore, seems that a direct tangle measurement is currently resource in some quantum communication schemes. A mea- unfeasible; however, an indirect indication for entanglement, surement of the increased error would be an indirect indica- detected through gate error enhancement, is within experi- tion for the enhancement of entanglement by squeezing. mental reach. One important limitation for observing the error modifi- cation effects in free space is the spatial overlap between the ACKNOWLEDGMENTS atom and the paraxial pulse mode. Similar limitations make We appreciate useful discussions with Ilya Averbukh, up a major difficulty in observing the effect of the inhibition Eran Ginossar, and Barak Dayan. R.O. acknowledges sup- of the atom coherences decay in a squeezed vacuum reser- port from the ISF Morasha program, the Chais family pro- ͓ ͔ voir, found by Gardiner 16 . In this context, other schemes, gram, and the Minerva foundation. involving atom-squeezed vacuum interaction in a cavity, were considered ͓37͔. Recent theoretical analysis and experi- ment, however, have shown a possibility of strong coupling APPENDIX: QUANTIZED PARAXIAL APPROXIMATION of a laser mode to a single atom in free space ͓38–40͔. The use of such beams, where the overlap between the incoming Here we would like to derive the model Hamiltonian of beam and the atomic dipole radiation pattern is large ͑ the atom-pulse mode interaction in free space seen in Eq. ͑2͒. ϳ⌫͒, may greatly enhance the effects discussed above. We start from the usual atom-field Hamiltonian in the dipole In many experiments, a pair of ground, e.g., hyperfine, approximation states of an atomic system are used as a qubit. Although the ˆ ˆ ˆ ˆ energy separation between ground states is typically in the H = HF + HA + HAF, radio-frequency range, quantum gates are often driven with a † laser beam pair, off resonance with a third optically excited ˆ ប ជ HF = ͚ kaˆ kជaˆ k, level, in a Raman configuration. The limitations to hyperfine kជ qubit coherence and gate fidelity due to the quantum nature of the Raman laser fields were investigated both experimen- 1 ͓ ͔ ˆ ប tally and theoretically 41,42 . In this case too, it is interest- HA = a ˆ z, ing to ask whether the use of, either independently or rela- 2 tively, squeezed light fields would influence the gate error. ជ ជ ប ik·ra An experiment, similar to that described above but with a k e ˆ ͱ ˆ ͩ ជ ͪ͑ ͒ H = ͚ i d aˆ − H.c. ˆ + ˆ , ground-state qubit could benefit from the ground-state qubit AF ⑀ k k ͱ + − ជ 2 0 V intrinsic longevity. In fact, a gate averaged error of few 10−3, k dominated by laser classical noise and photon scattering, was ͑A1͒ recently measured on a single trapped-ion hyperfine qubit where kជ are the indices of a Fourier mode and its polariza- and a Raman laser gate ͓43͔. tion, respectively, and rជa is the position of the atom. The To conclude, we have calculated the entanglement be- ជ tween a short pulse of resonant squeezed light and a two- dipole matrix element d is assumed to be real and dkˆ ͑ ͒ ជ ˆ ជ ជ ˆ level atom in free space during a qubit i ˆ y gate a pulse = k ·d, where k is the polarization vector with polariza- and the resulting gate error. Several differences arise when tion and is perpendicular to the propagation direction kជ. comparing to coherent control with CL. As could be ex- We first divide the sums on Fourier modes into the paraxial ⌳ pected for squeezed light, three phases determine the error subspace modes, p, and the nonparaxial subspace modes. magnitude—the squeezing phase, the light phase, and the The paraxial subspace modes include the modes centered ជ phase of the initial qubit superposition. In comparison with around some carrier wave vector in the z direction k0 =k0zជ: the coherent light case, the error and the entanglement can be ⌳ ͕⌳x ⌳y ⌳z ͖ either enhanced or suppressed depending on the above phase p = p, p, p ,
033803-9 SHAHMOON, LEVIT, AND OZERI PHYSICAL REVIEW A 80, 033803 ͑2009͒
⌳x ͑ co co͒ p = − kx ,kx , 0 1 ͑ ͒ ͱ i k0+k za ͑ ͒ gk = d e . A8 2⑀ បA ͱL ⌳y ͑ co co͒ 0 p = − ky ,ky , ˆ ͕ ͖ As for HF, since all the modes discussed here, q,i,k , are ⌳z ͑ co co͒ ͑ ͒ p = k0 − kz ,k0 + kz , A2 eigenmodes of free space in the paraxial approximation, they have well-defined single-particle energies ប , ប , and ប , where the bandwidths for ͑x,y,z͒ directions are narrow, i.e., q i k respectively. The total Hamiltonian from Eq. ͑A1͒ thus be- co Ӷ ki ko, comes ˆ ˆ ˆ ˆ i = x,y,z. ͑A3͒ H = HF + HA + HAF, The nonparaxial modes are simply the rest of the modes kជ ⌳ ͕ ͖ Hˆ = ͚ ប aˆ †aˆ + ͚ ប aˆ †aˆ + ͚ ប aˆ †aˆ , not included in p and we denote them with indices q . The F q q q i i i k k k ⌳ q i k paraxial subspace p can be approximately spanned by the modes that are the solutions of the paraxial wave equations ͑ ͒ k0+k 1 in free space with carrier frequencies k = k0 +k c, TEMnm . ˆ ប ͑ ͒ HA = a ˆ z, These modes are denoted by the functions Unmk rជ 2 ͑ ͒ ͑ ͒ i k0+k z ͑ជ͒ =unmk rជ e . Their Fourier coefficients are Unmk k so ͒ ͑ ͒ ͑͒† ء ͑ that their mode operators are ˆ ប HAF = i ͚ gjaˆ j − gj aˆ j ˆ + + ˆ − . A9 ء ͑ជ͒ ជ ⇒ ជ ͑ជ͒ j=͕q,i,k͖ aˆ nmk, = ͚ Unmk k aˆ k aˆ k = ͚ Unmk k aˆ nmk,. ជ⌳ nmk k p We would now like to describe a pulse mode with a Gaussian ͑A4͒ transverse profile composed of Gaussian beams of different frequencies around the carrier frequency 0. The pulse mode Note that the Fourier modes here, aˆ ជ, are only those in- k is then spanned by the ͕k͖ modes. We can choose a new cluded in the paraxial subspace. Using Eq. ͑A4͒ it is easy to orthonormal basis ͕f͖, which spans the same subspace as that show within the paraxial approximation that ͕ ͖ of the k basis and that includes a pulse mode f0 as one of ជ ជ eik·ra its members. Both ͕k͖ and ͕f͖ modes share a similar Gauss- ͱ ˆ ជ Ϸ ͱ ͑ជ ͒ ͚ kdkaˆ k ͱ 0 ͚ daˆ nmk,Unmk ra , ian xy profile; we, therefore, focus only on their z depen- ជ V k nmk, dence. We write the relation between the k and f mode func- ͑A5͒ tions as ͕ ͖ ͕ ikz where 0 =k0c. We denote all the nm 00,k, and nm e ⌽ ͑ ͒ ik0z ͑ ͒ ik0z ͑ ͒ ͑ ͒ =00,k,=2͖ modes by ͕i͖, where ͕͖=͕1,2͖ are the two f z = e f z = e ͚ f k , A10 ͱL polarizations. The rest of the modes, ͕nm=00,k,=1͖, have k ͑ ͒ Gaussian transverse profile and are denoted ͕k͖. We thus get i k0+k z /ͱ ⌽ ͑ ͒ where e L is a k mode while f z is an f mode. An ͑ ͒ f mode is thus defined by its envelope f z . In a similar͑͒† ء ͑ ͒ ប͑͒† ء ͑ ប ˆ HAF = i ͚ gqaˆ q − gqaˆ q ˆ + + ˆ − + i ͚ giaˆ i − gi aˆ i ˆ + q i way, the transformation for the field operators is
aˆ = ͚ ͑k͒aˆ . ͑A11͒ + ˆ ͒ + iប͚ ͩͱ 0 dU ͑rជ͒aˆ − h . cͪ͑ˆ + ˆ ͒, k f f − ⑀ ប 00k a k + − f k 2 0 ͑ ͒ The pulse mode is then characterized by the operator aˆ , A6 f0 whereas the Hilbert space of the two-level atom is spanned where d=d=1 and gi, gq are the atom-mode dipole couplings by the Pauli-spin matrices. The atom and the pulse mode ͕ ͖ ͕ ͖ for the modes i,q . The dipole couplings for the k modes make for the relevant bipartite system. The rest of the EM are seen explicitly in Eq. ͑A6͒, modes, aˆ , aˆ , and aˆ , interact with the bipartite system via f f0 i q 0 0 ͑ ͒ the atom and make up a reservoir. g = ͱ dU ͑rជ ͒ = ͱ du ͑rជ ͒ei k0+k za. k 2⑀ ប 00k a 2⑀ ប 00k a We now focus on the part of the Hamiltonian that includes 0 0 only the atom and the Gaussian paraxial modes ͕k͖, and their ͑ ͒ A7 interaction We note that u ͑rជ ͒ is just the overlap of the k mode with 00k a 1 ˆ p ប † ប ប ͑ ͒͑ ͒ the atom. This overlap includes some normalization of a H = ͚ kaˆ kaˆ k + a ˆ z + i ͚ gkaˆ k − H.c. ˆ + + ˆ − . quantization length in the beam direction, L, which should k 2 k eventually be taken to be infinity, and which is multiplied by ͑A12͒ some effective area of the mode, A, evaluated at the atom’s ͑ ͒ /ͱ position, i.e., u00k rជa =1 AL. Considering the narrow band Moving to the interaction picture in the rotating wave ap- ͑ ͒ of k values around k0, A is approximately independent of k. proximation, assuming no detuning a = 0 and using Eqs. ͑ ͒ ͑ ͒ ͑ ͒ The couplings gk are then written A8 , A10 , and A11 ,weget
033803-10 QUBIT COHERENT CONTROL AND ENTANGLEMENT WITH… PHYSICAL REVIEW A 80, 033803 ͑2009͒
Hˆ p = iប͚ ͓g ͑t͒aˆ ˆ − H.c.͔, ͑A13͒ Hˆ ϵ Hˆ f0+atom = iបg ͑t͓͒aˆ ˆ − H.c.͔, ͑A15͒ I f f + S I f0 f0 + f with 0 ͑ ͒ ͱ ik0za where we take the envelope of the pulse, f z , to be real g ͑t͒ = d e ͑z − ct͒. ͑A14͒ 0 f ⑀ ប f a 2 0 A and set za =0. Denoting all the rest the EM modes apart from ͕ ͖ The system Hamiltonian of the relevant bipartite system f0 with indices r we get the total Hamiltonian in the inter- comprised of the atom and the pulse mode is then action picture in Eq. ͑2͒.
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