Table Of ContentHybrid Quantum Systems with Collectively Coupled Spin States:
Suppression of Decoherence through Spectral Hole Burning
Dmitry O. Krimer,∗ Benedikt Hartl, and Stefan Rotter
Institute for Theoretical Physics, Vienna University of Technology (TU Wien),
Wiedner Hauptstraße 8-10/136, A–1040 Vienna, Austria, EU
Spin ensemble based hybrid quantum systems suffer from a significant degree of decoherence
resultingfromtheinhomogeneousbroadeningofthespintransitionfrequenciesintheensemble. We
demonstratethatthisstronglyrestrictivedrawbackcanbeovercomesimplybyburningtwonarrow
spectral holes in the spin spectral density at judiciously chosen frequencies. Using this procedure
we find an increase of the coherence time by more than an order of magnitude as compared to the
casewithoutholeburning. Ourfindingspavethewayforthepracticaluseofthesehybridquantum
systems for the processing of quantum information.
5
PACSnumbers: 42.50.Pq,42.50.Ct,42.50.Gy,32.30.-r
1
0
2
Hybrid quantum circuits that conflate the advantages spin. Asitturnsout,thedecoherenceresultantfromthis
l of different physical systems to achieve new device func- broadeningiscurrentlythemajorbottleneckforthepro-
u
J tionalities have recently shifted to the center of atten- cessingofquantuminformationinthesehybridquantum
tion [1]. This is largely because a new generation of systems. First attempts at resolving this problem have
4
1 experiments [2–11] lends encouraging plausibility to the meanwhile been put forward: On the one hand, it was
vision of using such hybrid device concepts to reliably shown that the decoherence is naturally suppressed for
] store and manipulate quantum information [12–17]. In very strong coupling when the spectral spin distribution
h
particular, the recent achievements in strongly coupling realized by the ensemble falls off sufficiently fast in its
p
large spin ensembles to superconducting microwave cav- tails. Signatures of this so-called “cavity protection ef-
-
t ities [2–6, 11] hold promise for combining many of the fect” [18, 19], have meanwhile also been observed exper-
n
a advantageous features of microwave technology with the imentally [11, 20]. To fully bring to bear the potential
u long spin coherence times found, e.g., in crystallographic of this effect requires, however, to go to very high val-
q defects of diamond. uesofthecouplingstrength,whicharepresentlydifficult
[
Whereas the collective coupling to a whole ensemble to reach experimentally. On the other hand, sophisti-
2 of spins is the key to reach the strong-coupling limit, the cated concepts for the spectral engineering of the spin
v ensemble generally comes with the downside of being in- density profile have been proposed [21, 22]. These con-
7 homogeneouslybroadened,i.e.,thetransitionfrequencies ceptsrely,however,onastrongmodificationoftheintrin-
8 betweendifferentspinlevelsareslightlydifferentforeach sically predefined density profile that is again very chal-
4
lenging to implement experimentally. In this Letter, we
3
present a method that circumvents the problems of both
0
approachesbybuildingonaveryelementaryconceptthat
.
1
requiresonlyaconsiderablyreducedexperimentaleffort.
0
Specifically, we demonstrate that the burning of two ju-
5
diciously placed spectral holes in the spin distribution
1
: suffices to drastically increase the coherence properties
v
of the hybrid spin-cavity system. From the viewpoint of
i
X quantumcontrolourapproachconstitutesanewandeffi-
cientstrategytostabilizeRabioscillationsinthestrong-
r
a coupling limit of cavity QED [23–25]. Suppressing the
detrimental influence of inhomogeneous broadening, as
suggested in our work, could also prove to be a key ele-
ment for the realization of ultra-narrow linewidth lasers
[26, 27].
FIG.1: (Coloronline)Sketchofthestudiedhybridquantum
To connect our theoretical work directly with the ex-
system: a synthetic diamond (black) containing a spin en-
semble (red arrows) coupled to a transmission-line resonator periment we will study in the following the recently
(curved gray line) confining the electromagnetic field to a implemented case of a superconducting microwave res-
small volume. onator strongly coupled to an ensemble of negatively
charged nitrogen-vacancy centers in a diamond (see
Fig. 1) [2, 3, 11, 20]. Our starting point is the Tavis-
CummingsHamiltonian((cid:126)=1)[28],whichdescribesthe
∗[email protected] dynamics of a single-mode cavity coupled to a spin en-
2
semble in the dipole and rotating-wave approximation, thespinensembleonthecavity,sothatthecavityampli-
tude at time t depends on all previous events τ < t. By
H = ωca†a+ 21(cid:88)N ωjσjz+i(cid:88)N (cid:2)gjσj−a†−gj∗σj+a(cid:3)− p[2e0r]foorrmbinygcaarLraypinlagceoutrtaanssfotarmtioonfatrhyistrVaonlstmerirsasieoqnuaatniaoln-
j j ysis [18, 19], the total rate of decoherence turns out to
i(cid:2)η(t)a†e−iωt−η(t)∗aeiωt(cid:3) . (1) be Γ ≈ κ+πΩ2ρ(ωs±Ω) in the limit of large coupling
strengths, Ω > Γ and γ → 0. The value of Γ is thus
Here σ+, σ−, σz are the Pauli operators associated with determined by the spin density ρ(ω), evaluated close to
j j j
theindividualspinsoffrequencyωj. Eachspiniscoupled the maxima of the two polaritonic peaks, ω = ωs ±Ω,
withastrengthgj tothesinglecavitymodeoffrequency split by the Rabi frequency ΩR ≈2Ω due to strong cou-
ω ,inwhichphotonsarecreatedandannihilatedthrough pling. Ourapproachisnowtotakethisrelationliterally,
c
the operators a† and a. The probing electromagnetic which is tantamount to saying that the decoherence rate
fieldinjectedintothecavityischaracterizedbyitscarrier Γ can be strongly suppressed by burning two spectral
frequency ω and by the amplitude η(t). holes into the spin distribution ρ(ω) right at these two
Next, we derive the semiclassical equations of motion positions, ωh =ωs±Ω, such that ρ(ωh)=0. The width
using the Holstein-Primakoff-approximation [29] (imply- of the holes ∆h should be very small, such as to remove
ing that the condition (cid:104)σz(cid:105) ≈ −1 always holds), the only a negligible fraction of the spins by the hole burn-
k
rotating-wave approximation and neglecting the dipole- ing. On the other hand, ∆h is limited from below by the
dipole interaction between spins. With these simplifica- spin dissipation rate, ∆h >γ.
tions,whicharewelljustifiedfortheexperiments[11,20] To demonstrate the efficiency of our approach ex-
operating at low input powers of an incoming signal, the plicitly, we first perform a stationary analysis [A˙(t) =
equations for A(t) ≡ (cid:104)a(t)(cid:105) and Bj(t) ≡ (cid:104)σj−(t)(cid:105) acquire B˙k(t) = 0] of the transmission T(ω) through the mi-
the following form (in the ω-rotating frame), crowaveresonatorasafunctionoftheprobingfrequency
ω. This quantity, which is directly accessible in the ex-
A˙(t)=−[κ+i(ω −ω)]A(t)+(cid:88)g B (t)−η(t), (2a) periment[11,20],providesdirectaccesstotheoccupation
c j j
amplitudeofthecavity[T(ω)∝A(ω)]. Assumingγ →0,
j
the transmission T(ω) acquires the following form,
B˙ (t)=−[γ+i(ω −ω)]B (t)−g A(t), (2b)
j j j j
iκ
T(ω)= . (3)
where κ, γ are the dissipative cavity and spin losses, re- ω−ω −Ω2δ(ω)+i[κ+πΩ2ρ(ω)]
c
spectively.
Large spin ensembles (N ∼ 1012 in [11, 20]) are This expression is normalized such as to reach the maxi-
best described by the continuum limit of the normal- mum possible value max(|T(ω)|)=1 for suitably chosen
ized spectral density ρ(ω) = (cid:80)Ng2δ(ω−ω )/Ω2. Here ω, κ, and ρ(ω). The real function δ(ω) is the nonlinear
j j j
Ω = ((cid:80)Ng2)1/2 is an effective coupling strength which Lamb shift [30] defined as
j j √
is enhanced by a factor of N as compared to a single (cid:90) ∞ dω˜ρ(ω˜)
δ(ω)=P , (4)
couplingstrength,g ,sothatΩcanreachthevaluesnec-
j ω−ω˜
0
essary for the realization of the strong coupling regime.
The inhomogeneous broadening of the spin frequencies where P stands for the Cauchy principal value. In the
ω and coupling strengths g then lead to a finite-width reference case taken from the experiment [11, 20], ρ(ω)
j j
distribution ρ(ω) centered around a certain mean fre- has no holes, see Fig. 2(a), and the transmission |T(ω)|2
quency ω . The specific shape of this spectral density displays the well-resolved double-peak structure typical
s
ρ(ω) can typically be determined by a careful compari- for the strong-coupling regime, see Fig. 2(b). If we now
son with the experiment based on stationary [3] or dy- burn two narrow holes into the spin density at the rele-
namical [11] transmission measurements. In the follow- vant positions ω = ω ±Ω, see Fig. 2(d), and reevalu-
h s
ing we will use the same parameters as in [11, 20] taking ate |T(ω)|2 we observe a more than fiftyfold increase in
a q-Gaussian distribution [3] for ρ(ω) centered around thecorrespondingtransmissionpeakvalues,seeFig.2(e).
ω /2π = 2.6915GHz, a full-width at half-maximum of Thisdramaticchangeisallthemoresurprisingconsider-
s
γ /2π = 9.44MHz and a q-parameter of 1.39. The ing that the relative number of spins removed from ρ(ω)
q
cavity decay rate, κ/2π = 0.4MHz (half-width at half- through the hole burning is less than 3%.
maximum)andthecouplingstrengthΩ/2π =8.56MHz. To understand this behavior it is best to analyze the
The starting point for our strategy is the insight that real and imaginary parts of the denominator of T(ω),
the non-Markovian dynamics of the spin system, which see Eq. (3). For the observed transmission resonances
is described by ρ(ω) and strongly coupled to the cavity at ω = ω with a maximum value of T(ω ) = 1 to
r r
mode,canbeaccuratelymodeledbyanintegralVolterra occur, two conditions are satisfied simultaneously: (i)
equation for the cavity amplitude A(t) (see Eq. (5) be- (ω −ω )/Ω2 = δ(ω ) and (ii) ρ(ω ) = 0. Consider first
r c r r
low and [11, 20]). The latter includes a memory-kernel, condition (i): In the reference case without holes, see
which is responsible for the non-Markovian feedback of Fig. 2(c), the nonlinear Lamb shift δ(ω) displays rather
3
300 300
(a) (d)
)
ω
(
ρ
0 0
0.02 1
(b) (e)
2
)|
ω
(
T
|
0 0
400 (c) 400(f)
)
ω 0 0
(
δ
-400 -400
2.67 2.68 2.69 2.7 2.71 2.67 2.68 2.69 2.7 2.71
ω/2π [GHz] ω/2π [GHz]
FIG. 2: (Color online) Comparison of the cavity coupled to
the inhomogeneously broadened spin ensemble without and
with hole burning in the spin density profile (left and right
panels, respectively). Top row: The q-Gaussian spin density
distribution, ρ(ω), without and with hole burning at ω =
h
ω ±Ω. Bothholesareofequalwidth,∆ /2π=0.7MHz,and
s h
have a Fermi-Dirac profile. Middle row: Transmission T(ω)
without and with hole burning in ρ(ω) (note different y-axes
scale). Bottomrow: ThecorrespondingnonlinearLambshift
δ(ω). Filledcircleslabelresonancevaluesω ofthetransmis-
r
sion T(ω) occurring at the intersections between the Lamb
shift δ(ω) and the dashed line (ω−ω )/Ω2. At empty circles
c
such intersections are non-resonant (see text).
smoothvariationsinthevicinityoftheresonantfrequen-
FIG.3: (Coloronline)TransmissionthroughthecavityT(ω)
cies ωr, determined by the intersection of δ(ω) and a versus probe frequency ω for different locations of the holes,
straight line (ω−ωc)/Ω2. In contrast, for the case with ωh,inthespindensityprofile,ρ(ω)(thewidthoftheholesis
holeburning,seeFig.2(f),δ(ω)exhibitsrapidvariations ∆ /2π = 0.7MHz). (a) Red (gray) curve: |T(ω)|2 in lin-log
h
around the two resonance points within a very narrow scale versus ω for ω = ω ±Ω. Black curve: Transmission
h s
spectral interval. As a consequence, the resultant trans- in the absence of hole burning. (b) Yellow (light gray) areas
mission peaks become substantially sharper. Due to the markthemostpronouncedpeaksin|T(ω)|2inthepresenceof
hole burning. Blue (gray) areas stand for the secondary po-
second condition (ii) they also dramatically increase in
laritonic peaks which stem from the case without hole burn-
height. Note, that no resonance occurs at ω = ω be-
c ing. Dashed arrows designate the distance Ω between po-
cause ρ(ω) has a maximum at this point and condition R
laritonic peaks. The white vertical cut corresponds to the
(ii) is strongly violated, see Fig. 2(c),(f). A close ex-
transmission shown in (a).
amination of the structure of T(ω) shows, furthermore,
that the narrow transmission peaks resultant from the
hole burning do not replace the broad polaritonic peaks
between ω¯ = 0 and ω¯ = 16MHz: While for large hole
present in the reference case, but rather get to sit on top spacings (ω¯ (cid:38) 11.5MHz) the effect of holes is negligi-
ofthem,seeFig.3(a). Aswillbeseenbelow,thedifferent ble, in the interval 0.8MHz(cid:46)ω¯ (cid:46) 11.5MHz we always
resonance widths in T(ω) set two different time scales in
find two sharp peaks superimposed on the two polari-
the dynamics with, in particular, the sharp peaks in the
tonicpeaksapproximatelyattheholepositions. Closeto
transmission giving rise to an asymptotically slowly de-
ω =ω ±Ω these peaks are most pronounced and reach
h s
cayingdynamicswithastronglysuppresseddecoherence.
unity. Inthelimitwhentheholesareburntverycloseto-
gether(ω¯ (cid:46)0.8MHz)thesharppeaksmergeintoasingle
Toexplorewhetherthenarrowholesweburntintothe one, located directly at the central frequency ω with a
s
spectralspindistributionatω =ω ±Ωhave,indeed,the transmission maximum reaching again unity in the limit
h s
optimal location, we now also test all possible other hole of ω →ω [see the yellow cusp in Fig. 3(b)]. Using the
h s
positions symmetrically placed around the maximum of symmetry of ρ(ω) with respect to ω , this behavior can
s
ρ(ω) at ω = ω . In Fig. 3(b) we present the numeri- alsobeprovenanalytically(notshown). Tocheckthero-
s
cal results for T(ω) as a function of the probe frequency bustnessofourmethodwealsotesteddifferentfunctional
ω and of different hole locations ω = ω ±ω¯ scanned forms for the hole profiles (Fermi-Dirac, q-Gaussian and
h s
4
100 (a) without holes however, by a crossover to Rabi oscillations with a much
N(t) 10-2 slower asymptotic decay [see Fig. 4(b)]. Quite remark-
10-4 ably, the total decay rate Γ in this asymptotic time limit
100 (b) with holes can even be substantially smaller than the cavity decay
N(t) 1100--24 irdaetnetκifiaedlonaes.tThehismiisniamllatlhlyermeaocrheasbulreprviazliunegfsoinrcΓe κinwraes-
cent studies on the cavity protection effect [11, 19, 20].
2×10-5 (c) without holes Apparently a new type of physics is at work here: Al-
2A(t)| though the system is in the strong coupling regime, the
| 0 two spectral holes slow down the leakage of the energy
2×10-5 (d) with holes stored in the spin ensemble back into the cavity. In par-
2A(t)| ticular, when being even slower than the inverse of the
| cavity decay rate κ, this sets a new global time scale
0
0 500 1000 1500 2000 for Γ, corresponding to the width of the sharp resonance
t (ns)
peaks which we identified before in Fig. 3(a). From the
FIG. 4: (Color online) (a),(b): Decay of the cavity occu- mathematical point of view such a slow asymptotic be-
pation N(t) = (cid:104)1,↓|a†(t)a(t)|1,↓(cid:105) from the initial state, for havior is associated with the contribution of two poles
which a single photon with frequency ωc resides in the cav- in the Laplace transform of Eq. (5) [20], which appear
ity and all spins are unexcited. The asymptotic decay Ce−Γt when the holes in ρ(ω) reach a critical depth. The pole
with and without hole burning (see red lines) is determined
contributions also stabilize the long-time behavior when
by the constants Γ/2π = 3MHz in (a) and a drastically re-
theholesareshiftedawayfromthepolaritonicpeaks[see
ducedΓ=0.42κ=2π·0.17MHzin(b). (c),(d): Dynamicsof
Fig.3(b)], buttheoptimalholepositionsremaincloseto
|A(t)|2 under the action of eleven successive rectangular mi-
the polaritonic peaks. Note that despite the consider-
crowavepulsesofdurationcorrespondingtotheRabiperiod,
τ =2π/Ω =52ns, phase-switched by π (every second pulse able photon loss (N(t) (cid:28) 1) for long times the phase
R
is shown as a vertical gray bar). Also here the asymptotic coherenceisverywellpreservedhere,aclearsignatureof
decay is much slower due to the presence of the holes. In all which is the stable form of the Rabi oscillations. In this
panels the holes in ρ(ω) have a width ∆ /2π=1.4MHz and way a high “visibility” can be achieved, as required for
h
are burnt at t=0 at ωh =ωs±Ω. the efficient processing of quantum information [31].
To demonstrate the efficiency of the hole burning ef-
fectalsoforquantumcontrolschemes, wepumpthecav-
rectangulardistributions)andfoundqualitativelysimilar
itybyasequenceofπ phase-switchedrectangularpulses,
results to the Fermi-Dirac form employed for all of the
each with a duration corresponding to the Rabi period,
above figures.
τ = 2π/Ω and a carrier frequency ω = ω = ω . As
To reach our ultimate goal of understanding the influ- R c s
shown in [11], this procedure is very well suited to feed
enceofthespectralholeburningontheresultantdynam-
energyintothestronglycoupledcavity-spinsystem,lead-
ics, we now study the time evolution of A(t) explicitly
ingtogiantoscillationsofbothspinandcavityamplitude
for the resonant case ω = ω = ω . The expression for
c s
[seeleftpartsofFig.4(c,d)]. Notonlydoweobservethat
thecorrespondingVolterraequationcanbederivedfrom
thesedrivenoscillationsaremorepronouncedwhenburn-
Eqs. (2a, 2b) (see [20] for details),
ing holes at ω =ω ±Ω, but we find, in particular, that
h s
A˙(t)=−κA(t)− (5) the Rabi relaxation oscillations setting in after switch-
ing off the driving field are dramatically more long-lived
t
(cid:90) (cid:90)
than in the case without holes [compare right parts of
Ω2 dωρ(ω) dτe−i(ω−ωc−iγ)(t−τ)A(τ)−η(t).
Fig.4(c,d)]. Theseresultsconfirmtherobustnessaswell
0 as the general applicability of our approach for various
coherent-control schemes in the strong-coupling regime
To prove that our predictions are valid not only in the
of cavity QED.
semiclassical but also in the quantum case, we consider
the case when all spins are initially in the ground state In summary, we present an efficient method to sup-
and the cavity mode a contains initially a single pho- press the decoherence in a single-mode cavity strongly
ton, |1,↓(cid:105). It can be shown that the probability for a coupled to an inhomogeneously broadened spin ensem-
photon to reside in the cavity at time t > 0, N(t) = ble. By burning narrow spectral holes in the spin den-
(cid:104)1,↓|a†(t)a(t)|1,↓(cid:105), reduces to N(t)=|(cid:104)0,↓|a(t)|1,↓(cid:105)|2 = sity at judiciously chosen positions the total decay rate
|A(t)|2,whereA(t)isthesolutionofEq.(5)withtheini- is dramatically decreased to values that may even lie be-
tialconditionA(t=0)=1(externaldriveη(t)=0). For low the dissipation rate of the bare cavity. Experimen-
thecasewithoutholeburningthissolutionisrepresented tally, our approach can be implemented by exposing the
by the damped Rabi oscillations [see Fig. 4(a)] found al- cavity to high-intensity microwave signals with spectral
ready previously [11, 20]. By burning narrow holes in components near the desired hole positions. Due to the
ρ(ω) at ω =ω ±Ω (immediately before t=0), we ob- strong driving the spins at these frequencies will equally
h s
serve very similar transient dynamics, which is followed, populate their ground and excited state and will thus be
5
effectively removed from the coupling process with the other spectral hole burning technique as studied in [21].
cavity. Support by the Austrian Science Fund (FWF) through
Project No. F49-P10 (SFB NextLite) is gratefully ac-
knowledged.
I. ACKNOWLEDGEMENTS
We thank S. Putz, R. Amsu¨ss and J. Majer for sug-
gesting us to work on this topic and for pointing out
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