Probing Correlations, Indistinguishability and Entanglement in Microwave Two-Photon Interference

2.1 MB·English

by  C. Lang

Probing Correlations, Indistinguishability and Entanglement in
Microwave Two-Photon Interference
C. Lang,1 C. Eichler,1 L. Steffen,1 J. M. Fink,1,∗ M. J. Woolley,2,† A. Blais,2 and A. Wallraff1
1Department of Physics, ETH Zurich, 8093 Zurich, Switzerland.
2D´epartement de Physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, J1K 2R1 Canada.
(Dated: January 15, 2013)
PACSnumbers: 85.25.-j,42.82.Bq,42.50.Ar,03.67.Lx
Interference at a beam splitter reveals both indistinguishable microwave photons emitted from in-
classical and quantum properties of electromag- dependent triggered sources realized in superconducting
netic radiation. When two indistinguishable sin- circuits. In each source A(B) a transmon-type qubit is
3
1 gle photons impinge at the two inputs of a beam strongly coupled with rate g/2π = 169(177)MHz to a
0 splitter they coalesce into a pair of photons ap- transmission line resonator, see Fig. 1a. To create a sin-
2 pearing in either one of its two outputs. This gle photon using one of the sources, we coherently excite

n effect is due to the bosonic nature of photons and the qubit into a state α|g(cid:105)+β|e(cid:105) using a resonant mi-
a was first experimentally observed by Hong, Ou, crowave pulse. Then, we swap the qubit state into the
J and Mandel (HOM) [1]. Here, we present the resonatormodeAˆ(Bˆ)bytuningthequbittransitionfre-

8 observation of the HOM effect with two indepen- quencyforhalfavacuum-Rabiperiodintoresonancewith
1
dent single-photon sources in the microwave fre- the resonator. This creates the state α|0(cid:105)+β|1(cid:105) in the

quency domain. We probe the indistinguishabil- resonator,whichforβ =1correspondstoasingle-photon
]
h ity of single photons, created with a controllable Fock state [22]. The photon then decays exponentially
p
delay, in time-resolved second-order cross- and with Lorentzian spectrum through the strongly coupled
-
t auto-correlation function measurements. Using output port of the resonator into the input mode aˆ(cid:48)(ˆb(cid:48))
n
quadratureamplitudedetectionweareabletore- of the beam splitter at a rate κ/2π = 4.1(4.6)MHz, see
a
u solve different photon numbers and detect coher- Fig.1. Thetwophotonstheninterfereatthebeamsplit-
q ence in and between the output arms. This mea- ter and are emitted into the output modes aˆ and ˆb, see
[ surement scheme allows us to observe the HOM Fig. 1a. Using the dispersive interaction between qubit

1 effect and, in addition, to fully characterize the and resonator, we tune the emission frequencies of the
v two-modeentanglementofthespatiallyseparated two sources to an identical value of ν = 7.2506GHz.
r
8 beam splitter output modes. Our experiments This is achieved by adjusting the qubit transition fre-
5
constitute a first step towards using two-photon quencies ν to 8.575(8.970)GHz with a static magnetic
4 a
interference at microwave frequencies for quan- flux applied to the SQUID-loop of each qubit. For our
4
. tum communication and information processing, experiments, we sequentially create 20 single photons in
1
e.g. for distributing entanglement between nodes each source at a rate 1/t = 1/512ns ∼ 1.95MHz in a
0 r
of a quantum network [2, 3] and for linear optics sequence repeated every 12.5µs.
3
1 quantum computation [4, 5].
To probe the photon statistics in the beam splitter
v: Presently, HOM two-photon interference has been output modes aˆ and ˆb we use two spatially separated
i demonstratedexclusivelyusingphotonsatopticalortele- heterodyne detection channels [21, 23], see dashed box
X
com wavelengths. Experiments were performed with in Fig. 1b. Each channel consists of a set of semicon-
r
a photons emitted from a single source using parametric ductor linear amplifiers, a microwave frequency mixer
downconversion[1], trappedions[3], atoms[6], quantum for downconversion to 25MHz, followed by an analog-
dots [7] and single molecules [8]. The HOM effect has to-digital converter. After further digital downconver-
also been observed with two independent sources [9–15] sion to d.c. and filtering, we extract the two quadra-
realizingindistinguishablesingle-photonstateswhichare ture amplitudes X (t) and P (t) corresponding to
a/b a/b
required as a resource in quantum networks or linear op- the complex envelope of the two time-dependent signals
tics quantum computation. Such experiments have also S (t) = X (t)+iP (t) [24]. In contrast to many
a/b a/b a/b
been performed using donor impurities as sources [16] other HOM experiments in which photons in the beam
including NV-centers in diamond [17, 18]. Furthermore, splitter outputs are detected by single-photon counters,
the HOM effect has been employed to create entangle- ourmeasurementofthecomplexenvelopeisintrinsically
ment between ions [19] in spatially separated traps, and photon-numberresolvingforaveragedmeasurementsand
torealizeacontrolled-NOTgateinasmall-scalephotonic allows us to measure coherences of the electromagnetic
network [20]. field. The complex envelopes S (t) are used to com-
a/b
Here, we demonstrate the HOM interference of two pute the statistics required to extract all relevant quan-
2
a a
aˆ
aˆ(cid:48) b
Aˆ
c
d
200µm 200µm
e
Bˆ
ˆb(cid:48) ˆb f
1mm
b
Figure 2. Second-order correlation function measure-
ments. Cross-correlationG(2)(τ)ofindividualsinglephoton
ab
sources displaying single photon antibunching, source A in
Figure 1. Sample and schematic. a, False color micro-
graphoftheHOMdevice. Atransmonqubit(blue,seeinset) panel a and B in panel b. a-e, G(2)(τ) when operating both
ab
controlled by a flux (purple) and charge gate line (orange) sources simultaneously with delay times δτ = 0,50,150ns
through which current or microwave pulses are applied, re- displaying the HOM effect and its dependence on the tem-
spectively, is coupled to each coplanar waveguide resonator poral photon overlap at the beam splitter. f, Second-order
(green) with fundamental mode Aˆ(Bˆ). The coupling rate of auto-correlationG(a2a)(τ)ofmodeaˆforδτ =0displayingtwo-
the input port of the asymmetric resonator is by a factor of photoncoalescence. Seemaintextfordetailsandadiscussion
103smallerthanattheoutputport. Eachresonatoroutputis oftheoffsetsindicatedintheupperrightcornerofeachpanel.
coupledtotheinputmodeaˆ(cid:48)(ˆb(cid:48))(inset)ofamicrowavebeam Blue dots are data and red lines are theory.
splitter(darkred)withtwooutputmodesaˆ(ˆb). b,Schematic
oftheHOMexperimentwithtwoindividualcavityQEDsys-
tems for photon generation, a beam splitter (BS), and linear
multaneouslygeneratetwoindistinguishablephotonside-
amplification and heterodyne detection for correlation func- √
allyrealizingatwo-modeentangledstate(|20(cid:105)+|02(cid:105))/ 2
tion measurements and full quantum state analysis [21].
at the beam splitter outputs. The measured cross-
correlationofthebeamsplitteroutputpowersisobserved
to vanish G(2)(τ) ≈ 0 for all τ between −t /2 and t /2,
tum correlations measured here [21, 24]. To analyze the ab r r
seecoloredregioninFig.2c. Therefore,weconcludethat
digitaldata,signalprocessingandstatisticalanalysisare
both microwave frequency photons coalesce at the beam
performedinreal-timeusingfieldprogrammablegatear-
splitter. ThisistheHOMeffectwithmicrowavephotons.
ray (FPGA) based electronics [22]. The noise added by
In our experiments the spatio-temporal coherence of the
thedetectionchainisfullycharacterizedbymeasuringits
single-photonstatesisgovernedbyresonatordecayalone,
statistical properties when the output modes aˆ and ˆb of
and shows no significant additional dephasing. This is
the beam splitter are left in the vacuum state with both
in stark contrast to many other experiments in which
sources idle [21, 24]. To verify the single-photon charac-
decoherence resulting in random frequency differences
ter of each source individually, we have measured their
between interfering photons causes finite correlations at
second-order cross-correlation functions G(2)(τ). These
ab τ ∼0 [6, 11, 13, 18]. At τ =ntr (n=±1,±2,...,±10),
display clear antibunching [22], see Fig. 2ab.
thepeakatintegernon-zeromultiplesofthephotongen-
Toinvestigatetwo-photonquantuminterference,wesi- eration period reflects the product of the power in each
3
output (cid:104)aˆ†aˆ(cid:105)(cid:104)ˆb†ˆb(cid:105) which we have normalized to one. photon states.
All measured second-order correlation functions are To distinguish between an equal mixture of the states
normalized by a common scaling factor and a relative |20(cid:105) and |02(cid:105), compatible with the observed correlations,
√
gain between the two amplification channels. A remain- and their coherent superposition (|20(cid:105) + |02(cid:105))/ 2, we
ing small offset subtracted from the reconstructed corre- fully characterize the two-photon states created in our
lation functions is indicated together with its standard HOM experiment by quantum state tomography. This
deviation(extractedfrommultiplemeasurements)inthe allows us to probe the entanglement generated between
upperrightcornerofeachpanelinFig.2. Themeasured the coalescing two-photon states in the two output ports
correlation functions are in good agreement with analyt- ofthebeam-splitter. Thecreatedstatesarealsoreferred
ical calculations [25] (solid lines in Fig. 2), taking into to as NOON states. In contrast to the NOON states
accountthecavitydecayratesκextractedfromindepen- of propagating photons investigated here, NOON states
dent measurements and a fixed detection bandwidth of have also been investigated in superconducting circuits
20MHz chosen to reject experimental noise outside the with photons localized in resonators [28, 29].
desired band. To perform full quantum state tomography on prop-
To explore the level of indistinguishability between agating photons, we record four-dimensional histograms
the two interfering photons we introduce a time delay of the measured field quadratures X , P , X , and P .
a a b b
δτ on the order of the photon decay time 1/κ ∼ 37ns. This is an extension of the scheme discussed in [21] to
For δτ = 50ns the correlation function G(2)(τ) remains two spatially separated modes. From these measure-
ab
close to zero at τ = 0, indicating the coalescence of ments we extract all moments of the two-mode field,
those photons detected with vanishingly small time dif- i.e. expectation values of the form (cid:104)(aˆ†)naˆm(ˆb†)kˆbl(cid:105) with
ference[6,25,26],seeFig.2d. Typicallythiseffectisdiffi- n,m,k,l ∈{0,1,2}. We observe that the first-order mo-
culttoobservewithdetectorsofinsufficientbandwidthor ments and all other odd-order moments are zero since
sources with significant dephasing rates [13]. The small all single-photon Fock states are characterized by a fixed
positive correlations observed at τ ∼ δτ are due to the photon number and consequently a fully random phase,
decreased temporal overlap of the single-photon spatio- see Fig. 3a. Since each mode carries exactly one pho-
temporalmodefunctionsatthebeamsplitter,i.e.thein- ton on average, (cid:104)aˆ†aˆ(cid:105) and (cid:104)ˆb†ˆb(cid:105) are close to unity, while
creaseddistinguishabilityofthetwophotons. Atτ ∼nt allothersecond-ordermomentsvanish. Thefourth-order
we observe broadened, lower amplitude correlations. Alrl moment (cid:104)aˆ†aˆˆb†ˆb(cid:105) is observed to be zero, while (cid:104)aˆ†aˆ†aˆaˆ(cid:105)
features are in agreement with theory (solid lines) [25]. and(cid:104)ˆb†ˆb†ˆbˆb(cid:105)areunity,consistentwiththecoalescenceof
For δτ = 150ns (and δτ = 100ns, not shown) the en- the two photons into either output. The above observa-
velopes of the two single-photon mode functions barely tions are consistent with the ones based on the correla-
overlap at the beam splitter, resulting in fully distin- tionfunctionmeasurementsinFig.2. Mostimportantly,
guishable single photons. At τ =±δτ and τ =nt ±δτ, the two-mode entanglement is indicated by the moment
r
see Fig. 2e, we observe positive correlations with an am- (cid:104)aˆaˆˆb†ˆb†(cid:105) which is close to 1, as expected. All measured
plitude 1/4, which originate from single photons imping- moments of the two-mode entangled state created in our
ingonthebeamsplitteratdifferenttimesandatdifferent HOM experiment are in good agreement with the pre-
input arms, compare Fig. 2ab. At τ = nt the correla- dicted ones, see wire frames in Fig. 3a. Note that mo-
r
tions between photons in the same beam splitter input mentsoforderfiveandhigherareallclosetozerowithin
arm sum up to 1/2, as expected. their statistical errors.
To clearly distinguish between single-photon anti- Inaddition,wehavedeterminedthemostlikelydensity
bunching and two-photon coalescence in time-resolved matrix ρ characterizing the created two-mode entangled
correlation function measurements, we have also mea- propagating photon state from the measured moments
sured the second-order auto-correlation function G(2)(τ) and their respective standard deviation following [21].
aa
Wehaverestrictedtheevaluationofmomentstolessthan
ofmodeaˆ. Whenoperatingonlyonesingle-photonsource
G(2)(0)isexpectedtovanish. However,intheHOMcon- three photons per output mode, since we create no more
aa
than two single photons with our sources. Note that a
figuration with δτ = 0 we find G(2)(0) = G(2)(nt ) =
aa aa r relatedanalysishasrecentlybeenperformedinRef.[30].
G(2)(nt ) = 1, see Fig. 2f, as there is a 50% probability
ab r TherealpartofρisshowninFig.3b, allelementsofthe
of detecting two photons in mode aˆ. All measurements
imaginary part of ρ are smaller than 0.02 (not shown).
of G(a2a)(τ) are in good agreement with calculations [25], WeextractafidelityoftheNOONtypestateof84%and
both for δτ = 0 (solid lines) and for δτ = 100ns (not a negativity of 0.39.
shown). Here,itisinterestingtonotethatasecond-order Finally, to explore the interplay between single- and
auto-correlationfunctionisrarelydirectlymeasured, be- two-photoninterference, wehaveperformedexperiments
cause of the thelack ofsufficientlyfast single-photon de- with modes Aˆ and Bˆ prepared in superpositions of
√
tectors [27]. In contrast, using our heterodyne detection 0 and 1 photon Fock states, i.e. (|0(cid:105) − i|1(cid:105))/ 2 and
√
scheme we are capable of measuring G(2)(τ) for multi- (|0(cid:105)+eiϕ|1(cid:105))/ 2 with variable phase ϕ, ideally creat-
aa
4
a 1st 2nd 3rd 4th b
0.5
0.25
0010200111 21 02 12 22221202211101201000
c 1st 2nd 3rd 4th d
Figure 3. Full quantum state tomography of propagating photon states created in HOM experiments. a, Mea-
√
suredmoments(cid:104)(aˆ†)naˆm(ˆb†)kˆbl(cid:105)withn,m,k,l∈{0,1,2}uptofourthorderandb,densitymatrix(realpart)for(|20(cid:105)+|02(cid:105))/ 2
with fidelity 84%. Colored bars are extracted from data, error bars indicate the standard deviation and wire frames represent
√ √
ideal state. c, Measured moments for ( 2|00(cid:105)+2|01(cid:105)+|20(cid:105)+|02(cid:105))/ 8 with corresponding state fidelity 84%. d, Power
detected in the two output modes ((cid:104)aˆ†aˆ(cid:105), blue) and ((cid:104)ˆb†ˆb(cid:105), red) for operating only source B (line) or both sources (dots)
creating superposition states with relative phase angle ϕ, see text for details.
√
ing the state ( 2|00(cid:105)−i(1−eiϕ)|10(cid:105)+(1+eiϕ)|01(cid:105)+ tum repeater or quantum communication applications
√
eiϕ(|20(cid:105)+|02(cid:105)))/ 8 at the beam splitter output. With based on microwave photons.
these input states, we have first measured the power in This work was supported by the European Re-
the beam splitter output modes aˆ (blue line) and ˆb (red search Council (ERC) through a Starting Grant and by
line)whenonlyoperatingsourceB andkeepingsourceA ETHZ. L.S. was supported by EU IP SOLID. A.B. and
idle, see Fig. 3d. In this case, we observe a power level M.J.W. were supported by NSERC, CIFAR, and the Al-
corresponding to 1/4 of a single photon independent of fred P. Sloan Foundation.
the phase angle ϕ, as expected for an equal superposi-
tionof|0(cid:105)and|1(cid:105)impingingonabalancedbeamsplitter.
OperatingbothsourcesAandB,weobserveasinusoidal
interferencewithphaseϕofthetwosuperpositionstates
in the beam splitter output power of mode aˆ (blue dots) ∗ Now at Institute for Quantum Information and Matter,
and ˆb (red dots), respectively. The sinusoidal oscillation California Institute of Technology, Pasadena, CA 91125,
USA
is a result of the interference between one photon in ei-
† Now at School of Engineering and Information Technol-
ther output port while the offset in power of 1/4 is the
ogy, UNSW Canberra, Canberra, ACT 2600, Australia
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