Beyond transcoherent states Field states for effecting optimal coherent rotations on single or multiple qubits

2025-05-06 1 0 1017.33KB 31 页 10玖币

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Beyond transcoherent states: Field states for

eﬀecting optimal coherent rotations on single or

multiple qubits

Aaron Z. Goldberg1,2, Aephraim M. Steinberg3,4, and Khabat Heshami1,2,5

1National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1N 5A2, Canada

2Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario, K1N 6N5 Canada

3Department of Physics and Centre for Quantum Information & Quantum Control, University of Toronto, Toronto,

Ontario, Canada M5S 1A7

4CIFAR, 661 University Ave., Toronto, Ontario M5G 1M1, Canada

5Institute for Quantum Science and Technology, Department of Physics and Astronomy, University of Calgary, Alberta

T2N 1N4, Canada

Semiclassically, laser pulses can be used to implement arbitrary transformations

on atomic systems; quantum mechanically, residual atom-ﬁeld entanglement spoils this

promise. Transcoherent states are ﬁeld states that ﬁx this problem in the fully quantized

regime by generating perfect coherence in an atom initially in its ground or excited state.

We extend this fully quantized paradigm in four directions: First, we introduce ﬁeld

states that transform an atom from its ground or excited state to any point on the Bloch

sphere without residual atom-ﬁeld entanglement. The best strong pulses for carrying

out rotations by angle θare are squeezed in photon-number variance by a factor of

sinc θ. Next, we investigate implementing rotation gates, showing that the optimal

Gaussian ﬁeld state for enacting a θpulse on an atom in an arbitrary, unknown initial

state is number squeezed by less: sinc θ

2. Third, we extend these investigations to ﬁelds

interacting with multiple atoms simultaneously, discovering once again that number

squeezing by π

2is optimal for enacting π

2pulses on all of the atoms simultaneously,

with small corrections on the order of the ratio of the number of atoms to the average

number of photons. Finally, we ﬁnd ﬁeld states that best perform arbitrary rotations

by θthrough nonlinear interactions involving m-photon absorption, where the same

optimal squeezing factor is found to be sinc θ. Backaction in a wide variety of atom-ﬁeld

interactions can thus be mitigated by squeezing the control ﬁelds by optimal amounts.

Contents

1 Introduction 2

1.1 Jaynes-Cummingsmodel................................. 3

2 Optimal ﬁeld states for generating arbitrary amounts of atomic coherence 4

2.1 Transcoherentstates ................................... 4

2.2 Beyondtranscoherentstates............................... 5

2.3 Discussion......................................... 7

3 Optimal ﬁeld states for generating Θpulses on arbitrary atomic states 9

3.1 Averaging ﬁdelity over all initial atomic states . . . . . . . . . . . . . . . . . . . . . 10

3.2 Averaging ﬁdelity over initial states with known azimuth . . . . . . . . . . . . . . . 11

3.3 Discussion......................................... 13

4 Generating π

2pulses for collections of atoms 14

Accepted in Quantum 2023-03-22, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.12167v2 [quant-ph] 23 Mar 2023

4.1 Optimal pulses for maximum coherence generation . . . . . . . . . . . . . . . . . . 15

4.2 Semiclassicallimit .................................... 17

4.3 Perfect pulses cannot be generated for the Tavis-Cummings interaction . . . . . . . 20

4.4 Discussion......................................... 22

5 Arbitrary coherence cannot be generated in the presence of nonzero detuning 22

6m-photon processes 23

6.1 Beyond the Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2 Transcoherent states and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Conclusions 25

A Averaging ﬁdelity of π

2pulses over θ, φ 29

B Averaging ﬁdelity over θ, φ for any rotation 29

C Averaging ﬁdelity over θfor known φand any rotation 30

1 Introduction

Coherence underlies quantum phenomena. Familiar from waves, coherence gives rise to inter-

ference eﬀects that power quantum systems to be diﬀerent from and more useful than everyday

objects. Quantum coherence enables computation [1], measurement [2], teleportation [3], and more,

making it an important resource to quantify [4,5,6,7]. Our ability to generate and transform

coherence is thus vital to the success of these ventures. Here, we show how to ideally transfer

arbitrary amounts of coherence from light to atomic systems.

Previous work found the ideal states of light for transferring maximal coherence to a single

atom: transcoherent states do this job and can be approximated by easier-to-generate squeezed

light in the appropriate limits [8]. These are important to the plethora of applications requiring

maximally coherent atomic states, such as quantum engines [9] and quantum state preparation

with quantum logic gates [10]. In other applications, arbitrary superpositions of a two-level atom’s

ground and excited states |giand |eimay be desired, with the most general state being

|θ, φi ≡ cos θ

2|gi+ sin θ

2eiφ|ei;(1)

the atom may be a physical atom or any other physical system with two energy levels, known as

a qubit. We ﬁnd the ideal states for all of these applications and demonstrate how they avoid

residual light-atom entanglement and other deleterious eﬀects that ruin the quality of the atomic

coherence.

Light is routinely used for controlling atomic states. Strong, classical light with a frequency

close to the transition frequency of a two-level atom induces “Rabi ﬂopping” that coherently drives

the atom between |giand |eiat the Rabi frequency Ω0√¯n, where Ω0is the single-excitation Rabi

frequency [sometimes known as the vacuum Rabi frequency, taking values from kHz to tens of MHz

in atomic systems, depending on cavity parameters, and up to hundreds of MHz for circuit quantum

electrodynamics (QED) systems] and ¯nis equal to the intensity of the ﬁeld in the appropriate units

that amount to the single-photon intensity when the ﬁeld is quantized [11]. Waiting an appropriate

time Ω0√¯nt =θ, for example, will lead to an atom in state |girotating to state |θ, 0i. However,

even quasiclassical light in a coherent state is fundamentally made from a superposition of diﬀerent

numbers of photons [12,13], which each drive oscillations in the atom at a diﬀerent Rabi frequency;

these give rise to famous eﬀects such as the collapses and revivals of Rabi oscillations that help

demonstrate the existence of quantized photons underlying quasiclassical light [14,15].

When considering the quantized version of light’s interaction with a single atom, the Jaynes-

Cummings model (JCM) dictates that the light will generally become entangled with the atom

[16,17,18,19,20,21]. This prevents the atom from being in any pure state |θ, φiand always tends

Accepted in Quantum 2023-03-22, click title to verify. Published under CC-BY 4.0. 2

to degrade the quality of the atomic state thus created. Such is the problem that transcoherent

states surmount for θ=π

2and that we generalize here.

A natural, further generalization of these results is to ﬁeld states interacting with a collection

of atoms. While the dynamics between a single atom and a mode of light are straightforward

to solve through the JCM, the same with a collection of atoms, known as the Tavis-Cummings

model (TCM), cannot usually be done in closed form. This enriches our problem and allows us to

incorporate strategies from semiclassical quantization into our investigations.

The above interactions are linear in the electromagnetic ﬁeld operators. We lastly extend these

results for arbitrary atomic control to interactions that involve nonlinear contributions from the

electromagnetic ﬁeld, such as m-photon absorption processes. These showcase the reach of our

transcoherence idea well beyond the initial goal of transferring coherence from light to atoms.

1.1 Jaynes-Cummings model

The Jaynes-Cummings Hamiltonian governs the resonant interaction between a single bosonic

mode annihilated by aand a two-level atom with ground and excited states |giand |ei, respectively:

H=ωa†a+|eihe|+Ω0

2aσ++a†σ−,(2)

where ωis the resonance frequency (ranging from hundreds of THz for optical transitions in atoms

down to tens of GHz for transitions between Rydberg states to below ten GHz for hyperﬁne

transitions or superconducting qubits), σ+=σ†

−=|eihg|is the atomic raising operator, and we

employ units with ~= 1 throughout. The JCM characterizes light-matter interactions in a variety

of physical systems including circuit QED [22], cavity QED [23], and parametric ampliﬁcation

[24]. This interaction conserves total energy and total excitation number, as can be seen from its

eigenstates

|±, ni=|ni⊗|ei±|n+ 1i⊗|gi

√2,(3)

with eigenenergies ±Ωn

2for the quantized Rabi frequencies

Ωn= Ω0√n+ 1.(4)

These are responsible for the ﬁeld and the atom periodically exchanging an excitation with fre-

quency Ωn

2when the initial state is either |ni⊗|eior |n+ 1i⊗|gi. We will work in the interaction

picture with Hamiltonian

HI=Ω0

2aσ++a†σ−;(5)

the Schrödinger-picture results can thence be obtained with the substitutions |ni → e−iωnt |niand

|ei → e−iωt |ei.

When the atom is initially in its ground state and the ﬁeld in state Pnψn|ni, the evolved state

takes the form

|Ψ(t)i=ψ0|0i⊗|gi+∞

n=0

ψn+1 cos Ωnt

2|n+ 1i⊗|gi − isin Ωnt

2|ni⊗|ei

=∞

n=0 |ni ⊗ ψncos Ωn−1t

2|gi − iψn+1 sin Ωnt

2|ei.

(6)

Similarly, when the atom is initially in its excited state and the ﬁeld in state Pnψn|ni, the evolved

state takes the form

|Ψ(t)i=∞

n=0 |ni ⊗ ψncos Ωnt

2|ei − iψn−1sin Ωn−1t

2|gi,(7)

where we employ a slight abuse of notation whereby ψ−1= 0 because photon numbers must be

positive and Ω−1= 0 by extending the deﬁnition from Eq. (4).

Accepted in Quantum 2023-03-22, click title to verify. Published under CC-BY 4.0. 3

To achieve an arbitrary ﬁnal atomic state, the most intuitive procedure is to begin with the

atom in its ground state and the ﬁeld in the target atomic state cos θ

2|0i+isin θ

2eiφ|1iand wait

for the duration of a “single-excitation πpulse” Ω0t=πto enact the transformation

cos θ

2|0i+isin θ

2eiφ|1i⊗ |gi→|0i ⊗ cos θ

2|gi+ sin θ

2eiφ|ei=|0i⊗|θ, φi.(8)

We exhaustively show in Section 2how to achieve this transformation with other ﬁeld states, with

no residual atom-ﬁeld entanglement, at faster rates, and with more feasible pulses of light. Since

the free atomic evolution enacts φ→φ−ωt, we can generate the states |θ, φiwith any value

of φand simply allow free evolution to generate the same state with any other value of φ, so in

the following we set φ= 0 (alternatively, direct solutions with φ6= 0 can readily be obtained by

adjusting the relative phases between the photon-number states).

2 Optimal ﬁeld states for generating arbitrary amounts of atomic coher-

ence

What are the optimal ﬁeld states that can generate arbitrary pulse areas? That is, which ﬁeld

states

|ψi=X

ψn|ni(9)

can achieve the transformations

|ψi⊗|gi→|ψ0i⊗|θior |ψi⊗|ei→|ψ0i⊗|θi,(10)

where the former correspond to “θpulses,” the latter to “θ+πpulses,” and we have deﬁned the

atomic state |θi ≡ |θ, 0iby allowing θto extend to 2π? We speciﬁcally seek transformations for

which the ﬁnal state has zero residual entanglement between the atom and the light, such that the

atomic state can be used in arbitrary quantum information protocols without degradation.

2.1 Transcoherent states

In Ref. [8], we deﬁned transcoherent states as those enabling π

2pulses. For atoms initially in

their ground states, perfect π

2pulses can be achieved by the transcoherent states whose coeﬃcients

in the photon-number basis satisfy the recursion relation

ψn+1 =icos Ωn−1t

sin Ωnt

ψn(11)

to ensure that the amplitudes of |giand |eiin the evolved state in Eq. (6)are equal. This can

be satisﬁed by ﬁeld states with ψn= 0 for n>nmax for any chosen maximum photon number

nmax ≥1, so long as the total interaction time satisﬁes

Ωnmax−1t=π, (12)

which ensures that the highest-excitation subspace spanned by |±, nmax −1iundergoes a πpulse

such that it transfers all of its excitation probability from |nmaxi ⊗ |gito |nmax −1i ⊗ |ei. In

the large-nmax limit, these states strongly approximate Gaussian states with an average photon

number ¯nwhose photon-number distributions are squeezed from that of a canonical coherent state,

σ2

coh = ¯n, by a factor of π

Similarly, another set of transcoherent states has its photon-number distribution satisfy the

same recursion relation as Eq. (11), with the lowest-excitation manifold undergoing a (2k)πpulse

and the highest a (2k+ 1)πpulse for any k∈N0. This pulse, in the large-¯nlimit, corresponds

to a 4k+1

2πpulse produced by a coherent state with its photon-number distribution squeezed by a

factor of 4k+1

2π. Superpositions of such states with nonzero coeﬃcients all satisfying (2k)2nmax ≤

n≤(2k+ 1)2nmax will also enact perfect π

2pulses in a time Ωnmax−1t=π.

Accepted in Quantum 2023-03-22, click title to verify. Published under CC-BY 4.0. 4

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●π8

■2π8

◆3π8

▲4π8

▼5π8

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50 100 150 200

0.05

0.10

0.15

0.20

ψn

Figure 1: Photon-number probability distributions for ﬁeld states that exactly generate arbitrary rotations θ

on atoms initially in their ground states (various shapes correspond to diﬀerent values of θ). The ﬁeld states

are calculated using the recursion relation Eq. (15)with nmax = 200. Also plotted are the photon-number

distributions for coherent states with the same average energies (solid curves). For the same nmax and thus

the same value of Ω0t, a higher-energy pulse generates a larger rotation angle θ, with more photon-number

squeezing being necessary for larger rotation angles.

Another set of transcoherent states is found when the atom is initially in its excited state. This

setup requires the initial ﬁeld state’s coeﬃcients to satisfy a diﬀerent recursion relation to ensure

equal coeﬃcients of |giand |eiin Eq. (7):

ψn+1 =−isin Ωnt

cos Ωn+1t

ψn.(13)

As well, the lowest-excitation sector must now undergo a (2k+1)πpulse while the highest undergoes

a2(2k+ 1)πpulse. This happens in a time

Ωnmin t= (2k+ 1)π, (14)

corresponding in the large-¯nlimit to a 4k+3

2πpulse generated by a coherent state that has been

photon-number squeezed by 4k+3

2π. Superpositions of commensurate states will again achieve

perfect coherence transfer.

2.2 Beyond transcoherent states

We can generalize the recursion relations of Eqs. (11)and (13)to generate arbitrary atomic

states of the form of Eq. (1).

When the atom is initially in its ground state, it will evolve to a state of the form of Eq. (1)if

and only if the initial ﬁeld state’s coeﬃcients obey the recursion relation [again, c.f. Eq. (6)]

ψn+1 =itan θ

cos Ωn−1t

sin Ωnt

ψn.(15)

The same boundary conditions as for transcoherent states hold, meaning that we require interaction

times of the form of Eq. (12)such that the lowest-excitation manifold undergoes a 0πpulse and

the highest a πpulse; extensions to other excitation manifolds are similarly possible. We plot a

number of such states in Fig. 1.

When the atom is initially in its excited state, it will evolve to a state of the form of Eq. (1)if

and only if the initial ﬁeld state’s coeﬃcients obey the recursion relation [again, c.f. Eq. (7)]

ψn+1 =−itan θ

sin Ωnt

cos Ωn+1t

ψn.(16)

Accepted in Quantum 2023-03-22, click title to verify. Published under CC-BY 4.0. 5

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Beyondtranscoherentstates:FieldstatesforeectingoptimalcoherentrotationsonsingleormultiplequbitsAaronZ.Goldberg1,2,AephraimM.Steinberg3,4,andKhabatHeshami1,2,51NationalResearchCouncilofCanada,100SussexDrive,Ottawa,OntarioK1N5A2,Canada2DepartmentofPhysics,UniversityofOttawa,25TempletonStreet,Ottawa,O...

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