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**Magnetic strong coupling in a spin-photon system and**

**transition to classical regime**

### I. Chiorescu, N. Groll, Sylvain Bertaina, T. Mori, S. Miyashita

**To cite this version:**

### I. Chiorescu, N. Groll, Sylvain Bertaina, T. Mori, S. Miyashita. Magnetic strong coupling in a

### spin-photon system and transition to classical regime. Physical Review B: Condensed Matter and Materials

### Physics (1998-2015), American Physical Society, 2010, 82 (2), �10.1103/PhysRevB.82.024413�.

### �hal-01699804�

**Magnetic strong coupling in a spin-photon system and transition to classical regime**

I. Chiorescu,1_{N. Groll,}1_{S. Bertaina,}1,2_{T. Mori,}3,4_{and S. Miyashita}3,4

1* _{Department of Physics and The National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA}*
2

_{IM2NP-CNRS (UMR 6242), Université Aix-Marseille, 13397 Marseille Cedex, France}3* _{Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan}*
4

_{CRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan}共Received 8 April 2010; revised manuscript received 17 June 2010; published 14 July 2010兲

We study the energy level structure of the Tavis-Cumming model applied to an ensemble of independent
*magnetic spins s = 1*/2 coupled to a variable number of photons. Rabi splittings are calculated and their
*distribution is analyzed as a function of photon number n*_{max}*and spin system size N. A sharp transition in the*
*distribution of the Rabi frequency is found at n*_{max}*⬇N. The width of the Rabi frequency spectrum diverges as*

### 冑

*N at this point. For increased number of photons n*max

*⬎N, the Rabi frequencies converge to a value*propor-tional to

### 冑

*n*max. This behavior is interpreted as analogous to the classical spin-resonance mechanism where the photon is treated as a classical field and one resonance peak is expected. We also present experimental data demonstrating cooperative, magnetic strong coupling between a spin system and photons, measured at room temperature. This points toward quantum computing implementation with magnetic spins, using cavity quantum-electrodynamics techniques.

DOI:10.1103/PhysRevB.82.024413 PACS number共s兲: 42.50.Ct, 71.45.⫺d, 75.45.⫹j, 64.60.⫺i

**I. INTRODUCTION**

Interactions of quantum systems with electromagnetic
ex-citations are at the core of quantum information processing.
Using photons and photonic entanglement, qubits can be
de-tected and manipulated, and quantum information can, in
principle, be transferred over long distances.1 _{Of particular}
interest are the resonant modes in electromagnetic cavities,
which have the potential of inducing a strong coupling
re-gime such that the interaction outlast both photon’s decay
and qubit decoherence times.2,3 _{Following the work of}
Dicke4_{on multiatom superradiance, the case of a single atom}
*in interaction with n photons has been studied theoretically*
by Jaynes and Cummings,5_{and later on generalized}6–9_{for a}
*number of N otherwise noninteracting atomic systems. Other*
theoretical studies, applied to solid state systems,10_{have }
in-cluded environmental effects as well共e.g., in semiconducting
materials11,12_{兲 or ensemble-locking in a giant spin.}13 _{}
Experi-mentally, the phenomena of strong coupling regime has been
reached by using the electric field component of the
electro-magnetic excitations: in one or more atomic systems,14,15
semiconductors,16 _{and superconducting qubits in interaction}
with one17 _{or more photons.}18,19_{These studies prove the }
ap-pearance of the so-called vacuum-field Rabi splitting共VRS兲
in the absorbtion peak of a probing photon field.

In contrast, achieving large magnetic coupling between a
photon and a quantum spin, has been explored to a lesser
extent, due to the typical smallness of the magnetic
compo-nent *共B field兲 of the electromagnetic field. However, since*
spin-based qubits do reveal significant coherence times for
temperatures up to ambient value,20,21 _{the issue of coupling}
spin qubits to photons for data manipulation and transfer
becomes of increasing interest. In the usual magnetic
reso-nance methods, e.g., electron spin resoreso-nance共ESR兲, the
ab-sorption measurement of the electromagnetic field is related
to the energy structure of the spin system. Feedback effects
*of the B-field component on the spins are ignored which*

means that for a two-level system there is one absorbtion
peak at a frequency matching the levels separation. On the
other hand, as mentioned above in the case of electrical
cou-pling, photon absorption can probe the quantum mechanical
interaction between the quantum system and the cavity
pho-tons which leads to VRS. In this paper, we will study the
relation between these two cases, one classical and the other
one quantum, by comparing the effects of magnetic coupling
*between N noninteracting spins s = 1*/2 and the external
ra-diation field. The transition between the classical and
quan-tum case will be analyzed as well.

Because the wavelength of the external field is large com-pared with the distances between spins, all the spins interact with a single mode of electromagnetic field. In the classical case of spin resonance, the system Hamiltonian is given by

*H*S=*H*Z0+*H*acwhere*H*Z0is the Zeeman coupling to a static

*field Hz*
*H*Z0= −0*Hz*

### 兺

*i=1*

*N*

*mi*

*z*=ប0 2

### 兺

*i*

*N*

*i*

*z*, 共1兲

*where mz= −gs*

*B*

*i*

*z*_{/2 gives the magnetic moment of spin}

*i* *共gs*is the g-factor, equal to 2 for a free spin and*B* is the

Bohr magneton兲 and ប0 is the Zeeman splitting generated

*by Hz*. The term*H*acrepresents the spin coupling to an

*alter-nating B-field component h*_{0} oscillating with a frequency
/2. Using the notation ប⍀*R*=0*h*0*gs**B*/2 and the Pauli

projection and raising/lowering operators this term is written
as
*H*acR=
1
2ប⍀*R*

### 兺

_{i=1}*N*

*共eit*

_{}

*i*+

*+ e−it*

*i*−

_{兲}

_{共2兲}

*H*acX=ប⍀*R*cos共*t*兲

### 兺

*i=1*

*N*

*i*

*x*. 共3兲

When treating the radiation field quantum mechanically, as
in the Jaynes-Cummings model, the spin-photon coupling is
*described by a parameter g, here assumed to be the same for*
all spins

*H = H*Z0+*បg*

### 兺

*i*
*N*

*共bi*+*+ b*†*i*−兲 + ប*b*†*b* 共4兲

*with b, b*† _{the photon annihilation/creation operators. The}

amplitude of the electromagnetic field is a dynamical
vari-able but not an external parameter as in the case of Eq.共2兲 or
共3兲. Namely, the strength of the electromagnetic field is given
*by b*†*b in the quantum mechanical model in Eq.*共4兲 while it

*is given by h*0 in the classical models in Eqs.共2兲 and 共3兲.

**II. COMPLEX SUSCEPTIBILITY IN THE LINEAR**
**RESPONSE THEOREM**

In the linear response theory,22,23_{the imaginary part of the}
complex susceptibility is given by

### ⬙

共兲 =*1 − e*−ប 2

### 冕

_{−⬁}⬁

*具Mx共0兲Mx共t兲典*0

*e−itdt,*共5兲

*where Mx*= 1 2兺

*i*

*i*

*x*, 具¯典

_{0}= Tr

*¯e*−HZ0

*/Tre*−HZ0

_{,}

_{= 1}

*/k*B

*T,*

*k*_{B}*is the Boltzmann constant, and T is system’s temperature.*
This gives a coefficient of proportionality between the
in-duced quantity*具Mx典 and the field h*0*at h*0= 0. The

eigenval-ues of *H*Z0 *are simply given by Ek= kប*0*, k = −N ,¯ ,N,*

each of which is*NCk= N !/k!共N−k兲! times degenerate. There*

is only one energy difference*⌬E=ប*0that has nonzero

*ma-trix element of Mx*. In this case, each spin interacts only with

the field, individually. Thus, we have a single peak in

### ⬙

共兲 at=_{0}. If we include some interactions among spins, such as the dipole-dipole interactions, the degeneracy of the

*en-ergy levels of the N spin system would be resolved, and*additional peaks in⬘共兲 are expected.

**Quantum dynamics of paramagnetic spins under an ac field:**
**Rabi oscillation**

If we consider the dynamics of spins in the ESR
Hamil-tonian *H*_{S}, the total magnetization shows the so-called Rabi
oscillation. By transforming the wave function*兩⌽共t兲典 as*

*兩⌿共t兲典 = ei*共1/2兲_{0}*t*_{z}_{兩⌽共t兲典 ⬅ U兩⌽共t兲典}_{共6兲}

the rotating frame version of*H*Sis given by

*UH*S*U*−1=
1
2ប⍀*R*

### 兺

*i*

*N*共

*i*++

*i*−兲 = ប⍀

*R*

### 兺

*i*

*N*

*x*共7兲

*at resonance. In this representation, the z component of the*
*magnetization rotates around the x axis with the angular *
ve-locity *˙ =⍀R*, which is the Rabi oscillation. It should be

noted that the phase oscillates with the angular frequency0,

*which causes a rotation of the magnetization around the z*

axis in the laboratory frame. The effective eigenvalues of Hamiltonian共7兲 are given by

*E*
*˜*

*k= kប⍀R* *− Nⱕ k ⱕ N,* 共8兲

which are equidistant *共⌬E˜k*=ប⍀*R*兲 and again we consider

that each spin interacts only with the field, individually.

**III. QUANTUM TREATMENT OF SPIN-PHOTON**
**COUPLING**

Now we study the case where the interaction between spins and photons is treated quantum mechanically. We start by reviewing the energy diagram of model in Eq. 共4兲.

**A. Case of single spin N = 1**

*First, we consider the case of N = 1. We adopt the basis*

*兩n,*典 where n denotes the number of photons in the cavity

and= −/+ shows the ground/excited state as an eigenvalue
of*z*. The matrix of*H is separated into 2⫻2 blocks for each*
pair *兵兩n,−典,兩n−1,+典其, given by*

### 冢

−ប0 2*+ nប*

*បg*

### 冑

*n*

*បg*

### 冑

*n*ប0 2 +

*共n − 1兲ប*

### 冣

. 共9兲The eigenstates are given by

*E*_{⫾}=ប

### 冉

*n −*1 2

### 冊

⫾ ប ␦ 2, 兩⌿⫾典 = 1### 冑

2### 冉

### 冑

1⫿ ⌬/␦ ⫾### 冑

1⫾ ⌬/␦### 冊

, 共10兲where⌬=−_{0}and␦=

### 冑

⌬2*2*

_{+ 4ng}_{. At resonance}

_{}

_{=}

_{}0, the above reduces to

*E*

_{⫾}=

### 冉

*n −*1 2

### 冊

ប0*⫾ បg*

### 冑

*n,*兩⌿

_{⫾}典 =

_{冑}

1
2### 冉

1 ⫾1### 冊

. 共11兲Note that all blocks have the same photon number +
magne-tization constant,7_{C = n − 1}_{/2=共n−1兲+1/2.}

**B. Case of a spin ensemble N****⬎1**

*For N⬎1, the working basis becomes 兩n,兵*_{1}¯*N*其典

where*i*= −*/+ shows the ground/excited state of ith spin, as*

an eigenvalue of*i*
*z*

. When photons are absorbed or emitted
*by the spin ensemble, the quantity C = n + M is conserved,*6,7
*with M = m − N/2 the ensemble magnetization, m the number*
*of excited spins, and n the number of remaining photons.*
*The Hilbert space corresponding to all m values isNC*0+¯

+*NCN*= 2*N* in length. Further division in independent

sub-blocks can be done if one use a total spin representation.7 The system Hamiltonian can be written as

*CHIORESCU et al.* **PHYSICAL REVIEW B 82, 024413**共2010兲

*H =*ប0
2 *S*
*z*_{+}_{ប}* _{b}*†

*+*

_{b +}_{បg共bS}*†*

_{+ b}*−*

_{S}_{兲}

_{共12兲}

*with Sz,+,−*

_{=}

_{兺}

*i*

*N*

_{}

*i*

*z,+,−*_{. The Hamiltonian commutes with the}

total spin*共Sx*兲2+*共Sy*兲2+*共Sz*兲2and can be further separated into
*smaller blocks, classified by the total spin S. When all spins*
*are in the ground state, the total spin is maximum, S = N*/2.
*From here on we discuss the case S = N*/2 since further
pho-ton excitations and emissions will selectively couple states
within this subspace only. The subspace can be further
lim-ited if there are not enough photons to flip all the spins in the
*system: m ranges from 0 to min共N,n*max*兲 with n*max*= n + m the*

*number of photons for Sz= −N/2. Using a basis 兵兩m典,0ⱕm*

*ⱕmin共N,n*max*兲其 and the relation 共S=N/2兲*

*S*+_{兩S,M典 =}

### 冑

_{S}_{共S + 1兲 − M共M + 1兲兩S,M + 1典 =}### 冑

_{共m + 1兲共N − m兲}*⫻兩S,M + 1典* 共13兲

the diagonal and off-diagonal terms of Eq.共12兲 are

*Hm,m*=*共m − N/2兲ប*0+*共n*max*− m兲ប*,

*Hm,m+1*=*បg*

### 冑

*n*max

*− m*

### 冑

*共m + 1兲共N − m兲.*共14兲

At resonance, the diagonal term becomes *Hm,m*=*共n*max

*− N*/2兲ប_{0}*and is independent on m.*

**IV. EIGENVALUES FOR AN ENSEMBLE OF SPINS**

*For a total spin S = N/2, one expects N+1 eigenstates of*
*which N − n*max *are diagonal spin states and n*max+ 1 are

coupled spin-photon states *共if n*max*ⱖN, all N+1 states are*

coupled兲.

**A. Analytical expressions for n**_{max}= 0 – 3

*Analytical expressions for eigenvalues Em共n*max兲 are listed

below for few simple cases.

**1. Case n**_{max}= 0

In the vacuum-field where no photon exist when all spins are in the ground state, there is a unique state and its eigen-value is given by

*E*0共0兲= −

ប0*N*

2 . 共15兲

**2. Case n**_{max}= 1

In this case there are two coupled states:*兩n=1,m=N典 and*
*兩n=0,m=N−1典. The Hamiltonian of this block is*

### 冢

ប0### 冉

−*N*2

### 冊

+ប*g*

### 冑

*N*

*g*

### 冑

*N*ប0

### 冉

−*N*2 + 1

### 冊

### 冣

*兩n = 1,m = 0典*

*兩n = 0,m = 1典*=ប0

### 冉

−*N*2 + 1

### 冊

+### 冉

ប共−0*兲 g*

### 冑

*N*

*g*

### 冑

*N*0

### 冊

共16兲and the eigenenergies are given by *共m=0,1兲*

*Em*共1兲/ប =0

### 冉

−*N*2 + 1

### 冊

+ ⌬ 2 +### 冉

*m −*1 2

### 冊

### 冑

⌬ 2*2*

_{+ 4Ng}_{.}

_{共17兲}The VRS is given by

*E*

_{1}共1兲

*− E*

_{0}共1兲=ប

### 冑

⌬2*+ 4Ng*2 共18兲 and represents the rate at which the spin system coherently exchanges one photon with the radiation field.

**3. Case n****max= 2**

The spin-photon states span over three states: *兩n=2,m*
*= N典, 兩n=1,m=N−1典, and 兩n=0,m=N−2典. At resonance ⌬*
= 0, the Hamiltonian is given by

ប0

### 冉

2 −*N*2

### 冊

+### 冢

0*បg*

### 冑

*2N*0

*បg*

### 冑

*2N*0

*បg*

### 冑

2共N − 1兲 0*បg*

### 冑

2共N − 1兲 0### 冣

共19兲 and the eigenvalues are given by*共m=0,1,2兲*

*Em*共2兲/ប =0

### 冉

−*N*

2 + 2

### 冊

+*共m − 1兲g*

### 冑

*4N − 2.*共20兲 Here, the Rabi frequencies are degenerate

*E*_{2}共2兲*− E*_{1}共2兲*= E*共2兲_{1} *− E*_{0}共2兲=*បg*

### 冑

*4N − 2.*共21兲

**4. Case n****max= 3**

*Following a similar approach as for n*_{max}= 2, the
Hamil-tonian at resonance is given by

ប0

### 冉

3 −*N*2

### 冊

+## 冢

0*បg*

### 冑

*3N*0 0

*បg*

### 冑

*3N*0

*បg*

### 冑

4共N − 1兲 0 0*បg*

### 冑

4共N − 1兲 0*បg*

### 冑

3共N − 2兲 0 0*បg*

### 冑

3共N − 2兲 0## 冣

共22兲 with eigenvalues*Em*共3兲/ប = ប0

*共− N/2 + 3兲 ⫾ g*

### 冑

5共N − 1兲 ⫾### 冑

*共4N − 5兲*2

*+ 8N,*共23兲

*where m = 0, 1, 2, and 3, counts the four possible values.*

**B. General case N, n**_{max}ⱖ1

The size of the block representing the spin-photon states
*increases with n*max. For each photon made available to the

spin system, an additional spin will participate in the
coop-erative energy exchange and a new spin-photon state is
gen-erated, as indicated in Fig.1共a兲. This effect is exemplified in
Fig. 1, at resonance, for N = 5 where up to six states are
*generated with increasing n*max*. When n*max= 5, all spins are

*Although no new states are generated for values of n*_{max}
*⬎N, the N+1 eigenvalues will adjust to indicate the *
“over-saturation” with photons 共as shown with connected dots in
Fig.1兲. Analytically, this effect is shown by Eq. 共10兲 for N
*= 1 or by replacing N with n*_{max} in the off-diagonal terms
leading to Eqs. 共15兲, 共17兲, 共20兲, and 共23兲 and thus in the
corresponding Rabi splittings.

*For large values of n*max, all Rabi splittings

ប⍀*R= Em+1共n*max兲*− Em共n*max兲 共24兲

are equal to*⬇2បg*

### 冑

*n*max共as shown by vertical arrows in the

example of Fig.1兲. However, this transition to equidistance
is not a smooth process. In the following, we show that the
*spread in Rabi frequencies becomes maximal at n*_{max}*= N and*
that a gradual transition toward an equidistant spectrum
*de-velops for n*max*⬎N. Difference between consecutive *

eigen-values is shown in Fig.2. For N = 10关Fig.2共a兲兴 one observes
*a spread of Rabi splittings over few units of g for n*_{max}*= N*
= 10, followed by a collapse on a single-valued Rabi splitting
2បg

### 冑

*n*max关shown by continuous lines in Figs.2共a兲and2共b兲兴

*for n*_{max}larger than several tens.

*An exact diagonalization study for N up to 400 and three*

*n*max values provides support for a general view of the

pro-cess关Fig.2共b兲兴. At nmax*= N*共black dots兲 deviations from the

2បg

### 冑

*n*max limit 共black lines兲 are maximal and the range of

spread increases proportionally with

### 冑

*N. For n*max

*= N + 250*

*and N + 500 the distribution width of Rabi splittings is highly*
reduced. A convergence toward 2បg

### 冑

*n*maxis observed

*espe-cially at low values of N共or very large n*max*/N ratio兲, since in*

*this limit one approaches the analytical case of N = 1* 关Eq.
共11兲兴.

This convergence corresponds to the energy diagram of
the Rabi oscillation in Eq.共8兲 in the ESR model, where each
atom interacts with the field individually. The fact that ⍀*R*

⬀

### 冑

*n*

_{max}

*in this limit, resides on the dependence h*

_{0}⬀

### 冑

*n*

_{max}. If

*the number of photons is large enough, b, b*†

*→具b典, 具b*†典

*→*

### 冑

*n*

_{max}in Eq.共4

*兲. Thus, h*

_{0}in Eqs.共2兲 and 共3兲 corresponds to

### 冑

*n*max.

**C. Photon transmission spectra: a quantum to**
**classical transition**

*To probe the Em*
*n*

spin-photon states one could use a low
power beam and analyze the transmitted signal 共another
op-tion, demonstrated experimentally in the last secop-tion, is to
study the Fourier transform of the coherent emission of an
excited spin-cavity system兲. A low power probe could be
used to scan the frequency response of the cavity, after the
*introduction of n*_{max} photons of frequency 共=_{0} at
reso-nance兲.

A variable frequency beam, probing after the introduction
*of n*max photons, will see the Rabi splittings given by Eq.

共24兲. Therefore, in such experiment one should be able to
detect a transition in peak distribution from a large number
of ⍀*R* values to a single-valued transmission peak. By

*de-sign, such photon-driven transition cannot be observed at N*
= 1 but it would require at least several noninteracting spins
*coupled to n*_{max}*⬃N photons.*

*The passage of a system from n*max*to n*max+ 1 implies an

excitation from one group of eigenstates to the next one, as sketched in Fig.1共a兲. Due to the large number of eigenstates involved, the distribution of energy differences

0 1 2 3 4 5 6 7 8 9 10
-20g
0g
20g
40g
0
**(b)**
hω
>N
N
2
1
N=5
N+1
n_{max}> N
n_{max}< N
(E
(nmax
) -E
(0
) )/
h-n
max
ω
(n_{max})1/2
n_{max}= N
n_{max}+1
0
**(a)**
m

FIG. 1. 共a兲 For each additional photon, a new spin-photon state
*is generated, up to a number N + 1.*共b兲 Spin-photon eigenstates for

*N = 5, at resonance, obtained from diagonalization of blocks limited*

by *关min共n*max*, N兲+1兴, as sketched in the insert. For n*max*ⰇN, all*
*N + 1 states become equidistant, leading to equal Rabi splittings*

共shown by vertical arrows兲.

0 5 10 15 20
0g
25g
50g
n_{max}>>N
n_{max}>>N
n_{max}=N
m
(E
(nmax
) -E
(nmax
) )/
h
m+1
0 1 2 3 4 5 6 7 8 9 10
0g
10g
20g
m
**(b)**
N1/2
(n_{max})1/2
(E
(nmax
) -E
(nmax
) )/
h
n_{max}=N
**(a)**
m+1

FIG. 2. Rabi splittings calculated at resonance, as difference
between consecutive eigenvalues, for*共a兲 N=10 and 共b兲 N up to 400*
*and n*_{max}*= N共dots兲, N+250 共squares兲, N+500 共triangles兲; the N=1*
limit for Rabi splitting, 2*បg*

### 冑

*n*max, is shown by continuous lines. At

*n*

_{max}

*= N the distribution width in Rabi splittings is maximal and it*

*decreases rapidly with increasing n*max.

*CHIORESCU et al.* **PHYSICAL REVIEW B 82, 024413**共2010兲

ប⌬*m*⬘*m*
*共n*max兲_{= E}*m*⬘
*共n*max+1兲_{− E}*m*
*共n*max兲 共25兲
can be broad. As shown in Fig.3共a兲*for N = 10, the *
distribu-tion width is increasing as

### 冑

*n*

_{max}. However, only in the

*vi-cinity of n*max

*= N are the values highly dispersed, as shown*

*by a dense cloud of points at n*max*⬇N in Fig.*3共a兲. For large

*values of n*max, and in full agreement with the study of Fig.2,

the ⌬_{m}共n_{⬘}max* _{m}*兲

_{values are equally spaced by a Rabi splitting}2

*បg*

### 冑

*n*

_{max}.

A different probing method would consist in using the

*n*maxphotons to actually probe the energy levels. An

experi-mental demonstration for one superconducting qubit
*electri-cally coupled to n*_{max}= 0 , . . . , 5 photons has been recently
performed19 _{共see also Ref.}_{24} _{for a multiatom experiment}_{兲.}
In such a case, multiphoton transmission experiments can
*probe the energy difference between the Em*

*n*max* _{and E}*
0
0

_{关of Eq.}共15兲兴 ប=

*共Em*

*n*

_{max}

*− E*

_{0}0

*兲/n*max. 共26兲

*Above the transition threshold n*_{max}*⬇N, the photon *
fre-quency thus defined becomes gradually insensitive to the
dis-tribution in Rabi splittings due to the 1*/n*max factor. A level

diagram with Rabi splittings uniformly spaced by 2បg

### 冑

*n*max

will generate transmission peaks spaced by 2បg/

### 冑

*n*max. The

linear dependence on 1/

### 冑

*n*max

*for n*max

*ⰇN is shown in Fig.*

3共b兲, calculated at resonance conditions for N = 10. The

gradual passage from the quantum case to the classical ESR condition共single peak at=0兲 is visible with the increase

of the electromagnetic intensity.

**V. EXPERIMENTAL STUDY OF STRONG COUPLING IN**
**SPIN SYSTEMS**

We demonstrate the strong coupling between an ensemble
*of spins s = 1*/2 and photons using a sample of the
well-known ESR standard material,25 _{dipheriyl-picri-hydrazyl}
共DPPH兲. For an optimized cavity coupling, sample
position-ing, and size, we have been able to induce a sizeable Rabi
splitting observed for a series of Zeeman spin splittings_{0}.
A 1-s-long microwave pulse pumps a large number of
photons into a cylindrical cavity operated in mode TE011.

When the microwave is switched off, the cavity is coherently emitting photons corresponding to its own eigenmodes 共phe-nomenon known as cavity ringing兲. This ringing is detected by a homemade heterodyne analyzer. The experiment is per-formed at room temperature. To ease the distinction between the photons of the pump pulse and cavity’s own emitted photons after pumping, the pump is detuned by 50 MHz from cavity’s resonance /2= 9.624 GHz. The Fourier transform of the coherent oscillations is shown in Fig. 4共a兲 with the frequency axis shifted by /2 for clarity. The sample-loaded, no field 共0= 0兲 oscillation shows only the

cavity signature, located at /2.

For an applied magnetic field _{0}*Hz*=ប*/gs**B, with gs*

= 2, the spin system is in resonance with the cavity共0=兲.

The eight traces of Fig.4共a兲are for fields within ⫾0.4 mT
of the resonance condition. The Fourier transform of the
coupled spin-photon oscillations show peaks, indicated by
vertical marks in Fig. 4共a兲. The peaks location, relative to
cavity’s resonance/2, is shown in Fig.4共b兲, as a function
of the field detuning0␦*Hz*= −ប⌬/g*s**B*.

0.0 0.2 0.4 0.6 0.8 1.0
-2g
0g
2g
m
m'
0
Rabi splittings
(quantum)
n_{max}= N
N=10
(E
(nmax
) -E
(0) )/nh
-ω
1/(nmax)
1/2
single peak
(classical)
**(b)**
m
0 1 2 3 4 5 6 7
-100g
0g
100g **(a)**
N=10
(E
(nmax
+1
) -E
(nmax
) )/
h-ω
(n_{max})1/2
n_{max}= N

FIG. 3. *共a兲 Representation of all possible excitations from n*_{max}
*to n*max*+ 1, calculated at resonance, for N = 10. The corresponding*
*energy splittings are closely packed in the transition region n*_{max}
*⬇N and became equally spaced by 2បg*

### 冑

*n*max

*for n*max

*ⰇN. 共b兲*Mul-tiphoton, nonlinear resonances as a function of 1/

### 冑

*n*max. The linear

*dependence at high n*

_{max}indicates the emergence of an equally spaced Rabi spectra and the convergence to a single peak indicates the gradual passage to a classical ESR, single-peaked, resonance condition. -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 -10 -5 0 5 10

**(b)**DPPH: S=1/2, g=s 2

**cavity**

**emission**

**peaks**

**relative**

**to**ω

**/2**π

**(MHz)**

**detuning field (mT)**cavity resonanceω -20 -10 0 10 20 20 40 60 80 100 0 -0.4mT +0.3mT +0.2mT +0.1mT -0.1mT resonance -0.2mT

**Fourier**

**trans**

**form**

**o**

**f**

**cavity**

**emission**

**(arb.**

**units**

**)**

**F**ω

_{FFT}-**/2**π

**(MHz)**+0.4mT no field

**(a)**

FIG. 4. 共Color online兲 共a兲 Fourier transform of cavity ringing,
measured at room temperature for several magnetic field values
around the resonance condition. Frequencies are relative to the
sample-loaded cavity resonance, in absence of applied field. The
observed two peaks are indicated by vertical marks.共b兲 Peaks
po-sition as a function of detuning field0␦*Hz*. The dashed lines

indi-cate the classical level diagram, whereas the continuous line fit关Eq. 共27兲兴 shows a measured Rabi splitting of 10.9 MHz.

In a classical ESR experiment, for instance using the DPPH as a field calibration standard, one expects energy levels that follow the dashed lines in Fig.4共b兲. In particular, the cavity peak is visible and changes abruptly in size but not location, when the resonance condition is met 共at the inter-section of the dashed lines兲.

In our experimental conditions, due to the spin-photon
strong coupling, one observes two peaks separated by the
characteristic Rabi splitting. The size of the splitting can be
*attributed to the interaction between a group of N spins and a*
single photon, although the cavity contains a large number of

*n*max photons. A possible explanation resides on the

exis-tence, in standard ESR experiments, of the so-called “spin
*packets” characterized by a coherence time T*2and grouping

*an average number of spins N. The T*_{2}and cavity decay times
can be estimated from the peak widths of⬇6 MHz and 2.7
MHz at =0 共resonance兲 and 0= 0共no field兲 conditions,

*respectively: T*2= 170 ns and*cav*= 370 ns.

Consequently, the cavity photon depletion during
*emis-sion, marks the transition between the E*_{0,1}共1兲 *and E*_{0}共0兲levels of
Eqs. 共17兲 and 共15兲. Cavity ringing shows coherent
oscilla-tions with the frequencies*e0,1*/2, given by

*e0,1*−= −
⌬
2 ⫾
1
2

### 冑

⌬ 2_{+}

_{⍀}

*R*2

_{共27兲}

on which the continuous lines of Fig.4共b兲are based. The fit
procedure leads to a Rabi splitting of ⍀*R*/2= 10.9 MHz.

*We note that a detailed knowledge of the g and N parameters*
would require an on-chip type of experiment, at low
tem-peratures, to maximize the spin packet size and to ensure a
precise knowledge of spin position.26_{In a single photon }
pic-ture, and knowing the cavity volume 共⬃50 cm3_{兲, we }

esti-mate the number of spins contributing to ⍀*R* to be on the

order of 1020_{. Our data demonstrate the spin-photon strong}

coupling and also provide the mainframe to study experi-mentally the transition shown in Fig.3共b兲. The role of other factors in the spread of Rabi splittings, such as: dipolar or hyperfine interactions, local anisotropic crystal fields, and the size of the sample vs size of a cavity 共mode兲 can be studied as well. At the same time, the possibility to entangle spins and photons inside a cavity comes in strong support for the implementation of quantum computing algorithms by using magnetic spins and on-chip quantum electrodynamics meth-ods.

In conclusion, we present a study of cooperative
spin-photon interaction leading to a transition between a
quantum-type Rabi spectra to a classical ESR spectra. The
*transition requires a number of spins N*⬎1 and the
possibil-ity to gradually increase the number of photons. When the
*later is close to N, a dense Rabi spectra is numerically *
ob-served. For a large number of photons, the spectra gradually
become equidistant and the resonance peaks converge
to-ward the classical single-peaked, ESR resonance. The
theo-retical model is complimented by an experimental
demon-stration of the spin-photon strong coupling regime, which
*uses the B-field component of an electromagnetic mode in a*
cylindrical cavity.

*Note added. Recently, two similar experimental results*

have appeared.27,28

**ACKNOWLEDGMENTS**

This work was supported by the NSF Cooperative Grant No. DMR-0654118 and No. NHMFL-UCGP 5059, NSF Grant No. DMR-0645408, the Alfred P. Sloan Foundation, “Physics of new quantum phases in superclean materials” 共Grant No. 17071011兲, and also the Next Generation Super Computer Project, Nanoscience Program of MEXT. Numeri-cal Numeri-calculations were done on the supercomputer of ISSP.

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