Shannon Limits for LowTemperature Detector ReadoutKent D. Irwin

Citation: AIP Conference Proceedings 1185, 229 (2009); doi: 10.1063/1.3292320 View online: http://dx.doi.org/10.1063/1.3292320 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1185?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Recent Progress With Low Temperature Particle Detectors AIP Conf. Proc. 1185, 785 (2009); 10.1063/1.3292455 A Double Flux Locked Loop Scheme For SQUID Readout Of TES Detector Arrays Using The FDM Technique AIP Conf. Proc. 1185, 522 (2009); 10.1063/1.3292394 Application of LowTemperature Detectors to HighResolution Xray Spectroscopy AIP Conf. Proc. 1185, 419 (2009); 10.1063/1.3292367 Low Temperature Detectors: Principles and Applications AIP Conf. Proc. 1185, 3 (2009); 10.1063/1.3292362 Baseband Feedback for FrequencyDomainMultiplexed Readout of TES Xray Detectors AIP Conf. Proc. 1185, 261 (2009); 10.1063/1.3292328

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Shannon Limits for Low-Temperature Detector Readout

Kent D. Irwin

National Institute of Standard and Technology Boulder, CO 80305

Abstract. Information theory places constraints on multiplexing large low-temperature detector arrays. We discuss these constraints, and review the state of the art of multiplexed low-temperature detector arrays. The number of detectors read out by each output communication channel is much smaller than the theoretical maximum predicted by the Shannon-Hartley Theorem. We discuss emerging multiplexing techniques based on Walsh code-division multiplexing and hybrid multiplexing modulation functions that have the potential to increase the number of pixels multiplexed in each communication channel closer to the Shannon limit.

Keywords: SQUID, transition-edge sensor, microwave kinetic inductance detector, multiplexer. Shannon limit PACS: 85.25.Oj, 85.25.Pb, 87.19.lo

INTRODUCTION

Low-temperature detectors (LTD) are emerging as important tools for applications including astronomy, cosmology, materials analysis, particle physics, and security. Superconducting transition-edge sensors [1] (TES) have achieved the best x-ray [2] and gamma-ray [3] energy resolution of any non-dispersive technology. Both TES detectors and Microwave Kinetic Inductance Detectors [4] (MKIDs) can achieve background-limited performance in the CMB and submillimeter. This single-pixel performance has motivated the development over the last two decades of large arrays of low-temperature detectors. The largest LTD arrays now deployed are the kilopixel-scale TES instruments on the Atacama Cosmology Telescope (ACT) [5] and the South Pole Telescope (SPT) [6].

The development of kilopixel arrays has required the use of lithographic array microfabrication techniques to enable the integration of large numbers of pixels, and cryogenic signal multiplexing to reduce the number of wires required between temperature stages. Existing kilopixel TES arrays are read out with either Time-Division Multiplexing (TDM) or Frequency-Division Multiplexing (FDM). In TDM [7], each output charmel is used to read out multiple pixels by measuring their output signal sequentially. For instance, ACT has over 3,000 TES pixels read out using TDM with a multiplexing factor of 32:1 in 100 separate readout charmels. In FDM [8,4,9,10], each LTD is read out at a different frequency, and multiple

output signals are summed in each output chaimel. SPT has 960 TES pixels read out using FDM with a multiplexing factor of 8:1. SPT and ACT are surveying galaxy clusters using the Simyaev-Zel'dovich effect.

Existing arraying approaches used with LTDs have the potential to scale to arrays of ten thousand pixels, which would be sufficient for most CMB applications. However, many submilhmeter and x-ray apphcations would benefit from much larger arrays (hundreds of thousand to millions of pixels). Breakthroughs are necessary in both focal-plane engineering and cryogenic multiplexing to make it possible to scale to much larger pixel counts, and to reduce the cost and complexity of readout wiring even in smaller arrays.

While significant progress is needed in focal-plane engineering, including fabrication, absorber engineering, lead routing and thermal issues, in this paper we focus solely on cryogenic multiplexing. We start by considering the constraints on multiplexing large arrays of LTDs in a small number of output charmels from the perspective of information theory. We consider the state of the art in LTD arrays multiplexed with TDM and FDM, and describe techniques being developed to expand the scalability of LTD arrays and reduce the number of leads. These techniques include the introduction of multiplexing at microwave frequencies with superconducting microresonators [4,9,10], Walsh code-division multiplexing [II], and hybrid multiplexing basis sets [12-14]. Signals can also be multiplexed using the diffusion of either phonons [15,16] or quasiparticles

CP1185, Low Temperature Detectors LTD 13, Proceedings of the 13 International Workshop edited by B. Cabrera, A. Miller, and B. Young

2009 American Institute of Physics 978-0-7354-0751-0/09/$25.00

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[17,18]. However, diffusive multiplexing is beyond the scope of this paper.

INFORMATION CAPACITY OF CRYOGENIC AMPLIFIER CHANNELS

Information theory places a fundamental upper bound on the amount of information that can be transmitted in a communications channel such as a twisted pair on the output of a Superconducting Quantum Interference Device (SQUID) amplifier, or a coaxial cable on the output of a High Electron Mobility Transistor (HEMT). Information can be quantified in terms of its uncertainty, or entropy. A communication variable, such as the voltage on a wire, can take on a number of different distinguishable values depending on its dynamic range (the ratio of the maximum voltage to the rms voltage noise in a given bandwidth). If the communication variable is set to random values over its full dynamic range, the entropy is the logarithm of the number of accessible values. If a base two logarithm is used, the entropy is expressed as a number of bits.

The maximum number of bits per second that can be faithfully carried by an information channel is its carrying capacity, C. The Noisy Channel Coding Theorem [19,20] proves that if the information transmission rate is less than C, an ideal encoding scheme and error correction algorithm can reduce the error rate arbitrarily close to zero for sufficiently long blocks of information.

The Shannon-Hartley theorem [19,20] shows that, in the special case of a continuous-time channel with Gaussian noise, the information carrying capacity is

C B\og^{l + {SNRfy (1)

where B is the channel bandwidth and SNR is the rms signal-to-noise ratio assuming bandwidth B, expressed as an rms amplitude.

The fundamental "Shannon" limit on the number of LTD pixels that can be read out in a single communication channel is then

A „ C_ C,

(2)

where Cj is the Shannon bit rate of information carried on the analog output of a single low-temperature detector pixel. In practice, the number of pixels multiplexed will be well below A^ ^ unless

error correction algorithms are implemented. However, there is very large room for improvement

over the number of pixels presently multiplexed before the Shannon hmit is approached.

If the output noise of an LTD pixel is Gaussian, its channel capacity can be expressed as a Shannon-Hartley output bit rate using Eqn. I:

C, = B, log^ (l + {SNR,)'), where B, is the bandwidth

of the output of the single pixel and SNR, is the

signal-to-noise ratio of the LTD in bandwidth B,.

The Shannon-Hartley bit rate of different types of LTDs varies widely. For the purpose of comparing multiplexing schemes, we choose a fiducial LTD: a TES used as a ground-based 150 GHz CMB polarimeter [21]. The required pixel bandwidth is determined by the scan strategy that will be used, but a typical value is B, = 100 Hz. The incident photon power is about 5 pW, and the photon shot noise is about4x10" WHz- ' , giving SNR, «1.25x10" rms in bandwidth B,. From Eqn. I, the output Shannon-Hartley bit rate for this fiducial LTD is 2.7 kHz.

In most cases, a cryogenic amplifier is used to amplify the output signals of a LTD and transform its impedance to a level compatible with the wires between the base temperature and room temperature. The two most important cryogenic amplifiers are SQUIDs that are used to read out TESs and magnetic calorimeters, and HEMTs that are used to read out superconducting microresonators, including both MKIDs and microresonators coupled to TESs or magnetic calorimeters through microwave SQUIDs.

The channel capacity of a SQUID varies widely depending on the implementation. However, a typical SQUID has a maximum hnear input flux range of about AO « 0„ /;?•, where 0„ is the flux quantum. Typical SQUID input flux noise is 0„ «I|iO„/>//fe , in a bandwidth of a few MHz. The SNR can be increased significantly by feedback at low frequency, but the loop gain approaches unity at higher frequency. Using the conservative open-loop dynamic range, the SQUID Shannon-Hartley bit rate is of order 10 MHz.

The HEMT has a much higher bandwidth and dynamic range than a SQUID, but these parameters also vary widely. Assuming a cryogenic HEMT with saturation power -40 dBm and noise temperature of a few K in a 10 GHz bandwidth, the Shannon-Hartley bit rate is of order 100 GHz. Thus, much more information can potentially be carried on a HEMT-based communication channel. However, HEMTs dissipate much more power than SQUIDs and their coaxial cables conduct much more heat than the twisted pairs often used with SQUIDs.

From Eqn. 2, for our fiducial LTD, the Shannon-hmited number of pixels that can in principle be

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multiplexed per communication channel is of order 10 for SQUIDS and lO' for HEMTs. However, many fewer channels are presently multiplexed in practical instruments. In the following sections, we discuss the performance of different multiplexing approaches for cryogenic detectors, including how close they approach the Shannon limit, and consider advanced multiplexing techniques to make much more efficient use of the available information resources.

CRYOGENIC SIGNAL MULTIPLEXING

Analog signal multiplexing occurs in four basic steps: bandwidth limitation, encoding, summation, and demultiplexing/decoding. The first of these steps, bandwidth hmitation, is necessary to prevent signal degradation. If the bandwidth is not limited, high frequency detector noise and signal power will either alias into the signal band, degrading resolution as the square root of the number of pixels multiplexed, or appear in other channels as large crosstalk.

After bandwidth hmitation, the signals from different pixels are encoded by multiplying them with a set of orthogonal modulation functions and summing the modulated signals into the output channel [22]. The signals can be separated and decoded at room temperature using the same functions. The most important sets of orthogonal modulation functions used to date to multiplex TES detectors are low-duty-cycle square wave functions for time-division multiplexing (TDM) and sinusoidal functions for frequency-division multiplexing (FDM). We briefly review TDM and FDM, with which mature TES instruments have been deployed in the field. We discuss emerging techniques for FDM at microwave frequencies in superconducting microresonators for both MKID and TES detectors, and present Code-Division Multiplexing (CDM), which combines some advantages of both TDM and FDM. Finally, we describe hybrid basis sets, which have the potential to approach significantly closer to Shannon-limited channel capacity at microwave frequencies.

Time-Division Multiplexing

According to the Nyquist-Shannon sampling theorem [23], if the bandwidth of a signal is limited to df by a low-pass filter, a signal of duration dt can be exactly represented by Idfdt samples in time space. If the output communication channel has a bandwidth B:s> df , then many input channels can be multiplexed in one output channel by sampling them sequentially. In TDM, the dissipation in a resistive thermometer such as a TES makes it possible to

l ^ ^ ^ ^ ^

il :

0 0.5 1 1 ' ' '

I -0.5

-1 0.5

-1 C) 0.5 1

Time (frames) FIGURE 1. Time-Division Multiplexing Modulation Functions. The time (x-axis) is plotted against the output gain for the example of a four-pixel multiplexer (channels shown in four different colors). Orthogonal, low-duty-cycle square waves are used to turn on one channel at a time. All channels are cycled through sequentially each frame.

passively limit its bandwidth using L/R low-pass filters with relatively small inductors (typically a few hundred nanohenries for a TES with a normal resistance R ~10 vaD).

In TDM, the signal is thus encoded by multiplying it by a low-duty-cycle boxcar modulation function (Fig. 1). If the samphng interval, dt, is sufficiently short, no TES signal is ahased, and the input signal can be reconstructed at room temperature without loss. The bandwidth of the TES signal, df , can be reduced to below the Nyquist sampling frequency, Ijldt, but the bandwidth of the SQUID must be much larger than the Nyquist sampling frequency to accommodate all of the multiplexed signals. Thus, the effective noise current of the SQUID amplifier is degraded by SQUID noise ahasing. The minimum SQUID noise ahasing occurs if the bandwidth of the SQUID is itself limited by the same boxcar modulating function used to encode the TESs. In frequency space, the boxcar function is a sine function with noise bandwidth F^ = y2St,i„^, where (54„, = St/N is the time that the multiplexer dwells on each pixel, and A is the number of pixels multiplexed in each output channel. The SQUID noise bandwidth is thus A times larger than the Nyquist sampling frequency, and the SQUID noise is degraded by a factor of \[N by aliasing. This degradation factor is often referred to as the "multiplex disadvantage" of TDM. Thus, the SQUID must be coupled to the input channel with yjN higher mutual inductance than in the non-multiplexed case to avoid

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degradation of the input signal. This higher couphng increases the flux slew rate on the SQUID when a pulse is detected in a TES calorimeter.

Time-division multiplexing of TESs is typically implemented by connecting a separate SQUID to each TES, and summing the output into one communication channel [7]. The SQUIDs are turned on one at a time to implement the boxcar modulation function of Fig. 1. Multiple TES instmments based on TDM are now deployed or soon to be deployed in the field. A 3,000-pixel TES array is operating at the Atacama Cosmology Telescope [5] with 32 TES pixels in each output channel. Other astronomical instruments already deployed, or soon to be deployed with TDM include MUSTANG, GISMO, SPIDER, BICEP-2, the Keck Array, and ZEUS, and TDM is a leading candidate for the x-ray calorimeter array on the International X-Ray Observatory (IXO). TDM is also used in gamma-ray calorimeters for nuclear materials analysis [3], and x-ray calorimeters for synchrotron materials analysis, both of which consist of 256 pixels with a multiplexing factor of 32:1. The largest multiplexing factor with TDM is in the SCUBA-2 sub millimeter camera [24], with 40 TES pixels in each output channel, for a total of 10,000 pixels in the instrument. This multiplexing factor of 40 is a few hundred times smaller than the Shannon-hmited

multiplexing factor of SQUID-coupled TDM.

-10 calculated earher for

Frequency-Division Multiplexing

In frequency-division multiplexing (FDM) of an LTD array, the signals in different channels are modulated by sinusoids at different frequencies (Fig. 2). FDM is used for both TES and MKID arrays. In some implementations, FDM is conducted at MHz frequencies with SQUID amplifiers [8]. In other implementations, FDM is conducted at GHz frequencies with HEMT amplifiers coupled to superconducting microresonators. In the MKID, the microresonators are the sensing elements [4]. Microresonators can also be coupled through SQUIDs to TES pixels or magnetic calorimeters [10].

In FDM, unlike TDM, the amplifier can be limited to the same bandwidth as the LTD, and no amplifier noise aliasing occurs. Thus, FDM does not suffer from the multiplex disadvantage of TDM. This relaxes the requirements on the slew rate of the amplifier readout circuit. However, FDM requires significantly larger filter elements, and requires more complex room-temperature readout electronics than TDM.

0.5 1 Time (arb. units)

FIGURE 2. Frequency-Division Multiplexing Modulation Functions. The time (x-axis in arbitrary units) is plotted against the output gain for the example of a four-pixel multiplexer (channels shown in four different colors). The output of the four different pixels is modulated at four different orthogonal frequencies.

MHz Frequency-Division Multiplexing

For MHz FDM of TES circuits, the modulation function is implemented by ac biasing the TES itself [8,25] before amplification by the SQUID. Since the TES is a highly nonlinear device, ac biasing of TES detectors can introduce operational complexities [26]. The bandwidth of the TES is hmited by a lumped-element LCR bandpass filter. The R is provided by the TES resistance, R„~\D., with lumped-element inductors L~10 |aH and capacitors C~I nF. The amplitude of the carrier signal causes an unacceptably high flux slew rate in the SQUID, so the carrier must be nulled. In slow bolometric applications, the carrier nulling can be accomphshed by summing an amplitude- and phase-adjusted copy of the carrier into the SQUID feedback line. In fast x-ray applications, however, more complex room-temperature electronics are required that change the modulating envelope of the carrier-nulling signal during an x-ray pulse [22].

Multiple TES instruments are deployed or soon to be deployed using MHz FDM, including the SPT [6], APEX-SZ, and Polaibear FDM with baseband feedback is a candidate for the x-ray calorimeter array on the International X-Ray Observatory (IXO). Present instruments use multiplexing factors of 8:1, more than a factor of 1,000 smaller than the Shannon-limited multiplexing factor of -10"*. Plans are underway to increase this multiplexing factor to 32:1.

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GHz Frequency-Division Multiplexing

Cryogenic multiplexing circuits that operate at microwave frequencies with HEMT amplifiers have much larger Shannon limits on the number of LTDs that can be multiplexed in each communication channel. Frequency-division multiplexing of LTDs at microwave frequencies in tank circuits was first demonstrated with single-electron transistors [27]. Later demonstrations of multiplexing with MKIDs based on superconducting microresonators [4] showed great promise for scaling to large LTD arrays. The development of superconducting microresonators coupled to dissipationless SQUIDs [10] made it possible to multiplex TES or magnetic calorimeter arrays with these same large Shannon hmits.

In GHz FDM, multiple superconducting microresonators tuned to different frequencies are coupled to a single HEMT amplifier. A comb of microwave signals tuned to each resonator is reflected off of this array of resonators. When a signal is absorbed in the microresonator (in an MKID), or in a TES or magnetic calorimeter coupled to a particular microresonator through a dissipationless SQUID, the resonant frequency and/or Q of circuit is changed, which changes the reflected signal. The bandpass of the LTD is hmited by the superconducting microresonator whose bandwidth is determined at least partially by the value of a couphng capacitor to a 50 Q feedhne.

A prototype camera consisting of 16 two-color pixels (32 MKIDs) was recently tested at the Caltech Submillimeter Observatory [28]. In this initial prototype, only 4 pixels were operated at one time. A full-scale camera with 2,304 MKID channels is now being prepared. In this instrument, 144 MKIDs will be read out in each HEMT amplifier channel. This is a factor of more than 50,000 smaller than the Shannon limit. If the performance of available room-temperature electronics components increases sufficiently, it should be possible to read out a larger number of MKIDs in each HEMT. A practical limit may be set by the fact that the resonator center frequency varies by ~ 1 MHz because of uncontrolled variation in fabrication. If a decision is made not to risk accidental overlap of individual resonators, it would still be possible in principle to multiplex more than 1,000 MKID channels in each HEMT amplifier. Regardless, this multiplexing factor is still far below the Shannon limit of ~10'.

1-0 0. 11 '

0 0

^^•^B i[ 1 1 , 0 0 11 '

1 '

5 I

^^^^^^^^^^^H 5 1

' ^ ^ ^ ^

1 "I 5 1

1

0 0.5 1 Time (frames)

FIGURE 3. Code-Division Multiplexing Modulation Functions. The time (x-axis) is plotted against the output gain for the example of a four-pixel multiplexer (channels shown in four different colors). Orthogonal Walsh functions are used to modulate the polarity with which each pixel couples to the communications channel between positive and negative unity. A full modulation set is implemented in each frame.

Code-Division Multiplexing

A third set of orthogonal modulation functions is very promising for multiplexing TES detectors: Walsh matrices for code-division multiplexing (CDM) (Fig. 3). In code-division multiplexing, the signals from all TESs are summed with different polarity patterns. For instance, in the simplest case of two-channel code-division multiplexing, the sum of the signals from TES I and 2 would first be measured, followed by their difference. The original signals can be reconstructed from the reverse process. Depending on the implementation, CDM can use the same room-temperature electronics and firmware as TDM.

In CDM, like TDM, the bandwidth of the SQUID must be much larger than the Nyquist frequency associated with the samphng, so the SQUID noise is

degraded by a factor of y/Nby ahasing. However, CDM does not suffer from the multiplex disadvantage. Since every pixel is read out at all times (but with different polarities), A independent samples of each input signal are taken each frame, which increases the SNR by \1N . This increase in SNR compensates for the degradation due to SQUID noise aliasing.

CDM combines some of the attractive attributes of both time- and frequency-division multiplexing of TES detectors with SQUID amplifiers. Like time-division multiplexing, it uses simpler room-temperature electronics, smaller filtering elements, and

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it allows dc biasing of the TES sensor. Like frequency-division multiplexing, CDM does not suffer from the multiplex disadvantage, enabhng faster pulses and higher multiplexing factors.

The potential advantage of code-division multiplexing of TES detectors has previously been recognized [29,30], but implementations that have been proposed previously have not been pursued because of significant problems with energy resolution degradation and crosstalk. In these efforts, the Walsh matrix modulation function was implemented by changing the sign of the LTD bias, which made it impossible to hmit the TES bandwidth before multiplexing, leading to significant aliasing of detector noise.

Here we present two different approaches to code-division multiplexing that are both under active development at NIST for the readout of TES arrays [11]. These implementations hmit the bandwidth of the TES before multiplexing, so they do not suffer aliased TES noise. They promise improved array scahng and performance by eliminating the multiplex disadvantage of TDM while using its compact filter elements and simple room-temperature electronics. CDM with flux-summing is straightforward to develop, while CDM with current switches has greater potential scalability and performance.

Flux-Summing CDM

A conceptually simple way to implement code-division multiplexing is to passively sum A TES signal currents in A different superconducting transformers. The signals from the A transformers are then measured sequentially with a conventional time-division SQUID multiplexer. In the simplest case of two-channel code-division multiplexing, the sum of the signals from TES I and 2 would be measured in the first transformer, and their difference in the second. The original signals can be reconstructed from the reverse process.

Flux-summing CDM is straightforward to implement, but it requires leads to be lithographically routed from every TES to every SQUID, and it requires a superconducting transformer and time-division-multiplexed SQUID switch for each TES. In this approach, the orthogonal polarity code (typically a Walsh matrix) is implemented by the choice of the polarity with which the wire from each TES wraps around the different superconducting transformers. In contrast to the flux-actuated superconducting switch approach discussed next, the required length of lithographic leads scales as the square of the number of pixels multiplexed in each output channel. The complexity of lead routing in the flux summing

approach will provide a practical limit to how many total pixels can be multiplexed in each output channel. We are presently fabricating a 32-channel flux-summing CDM circuit that uses the same sihcon chip area as our conventional TDM circuits (3 mm x 20 mm).

While flux-summing CDM lacks some of the advantages of current-switch CDM, flux-summing CDM is much simpler to implement, and may provide near-term benefits for the multiplexing of fast x-ray calorimeters for applications including the International X-ray Observatory (IXO). Since CDM does not suffer from the multiplex disadvantage, the mutual inductance of the input coil can be significantly reduced, reducing the slew rate and dynamic range requirements of the SQUID. This improvement can increase the number of pixels multiplexed, or make it possible to multiplex faster detectors.

Current-Switch CDM

A more ambitious implementation of CDM uses a single-pole double-throw (SPDT) superconducting current switch to modulate the polarity with which the TES couples to the SQUID (Fig. 4). The polarity modulation occurs at much higher frequency than the bandwidth-hmited signal, so there is no degradation in performance from detector-noise aliasing.

The two circuit arms of the superconducting current switch contain superconducting-to-normal

t'eeJbdck 1 Feedback! Out 2

Code-division multiplexer

FIGURE 4. Circuit diagram for a 2x2 Code-Division Multiplexer based on a superconducting current-steering switch. When no current is passing through the address line associated with a row, the TES current flow through a branch of the circuit that couples with positive polarity to the SQUID. When an address current is applied, the current switches to a branch with negative polarity.

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switches based on a low inductance dc SQUID [31]. The superconducting-to-normal switch is a SQUID configured to operate as a variable resistor controlled by an applied flux. The critical current of a dc SQUID modulates from a value of /^ = 2/„ at 0 = 0 to a minimum value at O = 0„ /2 , where /„ is the critical current of a single junction.

The two arms of the SPDT circuit are actuated by an apphed flux, but the second arm has an additional n phase offset provided by 0„ /2 flux from a n phase line common to the entire array. Thus, when no flux is applied by the address line, switch 2 is "off (in the low critical current state), and switchl is "on" (in the high critical current state), and almost all of the TES current flows through switch 1. As the address flux is changed by 0„ /2 , switch 1 is driven into the resistive state, causing the signal to flow through switch 2.

We have conducted a 4-channel current-switch CDM demonstration [11], but no TES arrays using CDM have yet been fielded. If the performance goals of superconducting-switch CDM are achieved, it should be feasible to multiplex 256 TES pixels in each SQUID. This multiplexing factor would be about 40 times lower than the -10"* Shannon limit. While a significant improvement in multiplexing efficiency over TDM or FDM, it is still far from fundamental hmits.

Hybrid Multiplexing

In addition to TDM, FDM, and CDM, hybrid combinations of these modulation functions are possible. The combination of TDM and FDM [12], and the combination of two FDM functions [14] have been proposed in the past. The combination of TDM and FDM has been demonstrated [13]. In this demonstration, TDM SQUID switches were coupled to a set of GHz SQUID amplifiers that were themselves frequency-division multiplexed.

Multiplexing techniques using hybrid basis sets at microwave frequencies have the potential to operate closer to the Shannon limit. One promising combination is to use current-steering CDM switches [11] to multiplex a large number of TES detectors into dissipationless microwave SQUIDs coupled to superconducting microresonators [10]. It would be possible in principle for 256 TES pixels to be code-division multiplexed using current-steering switches into a single dissipationless microwave SQUID. One HEMT could in principle read out 256 dissipationless microwave SQUIDs each of which instruments 256 TES pixels in a CDM configuration. A series combination of GHz FDM and MHz TDM electronics could demultiplex the signals. In this way, it might be

TABLE 1. Estimated Shaimon Limits on readout of the Fiducial LTD (a 150 GHz polarimeter)

TDM MHz FDM GHz FDM CDM CDM + FDM

B

5MHz 5MHz

5 GHz

5MHz 5 GHz

Achieved (planned)

MUX factor 40

8(32)

4(144)(-1000)

(256) (65536)

Shannon Limit

-10" -10"

-10'

-10" -10'

possible to read out 65,536 TES detectors using one HEMT amplifier, about a factor of 100 from the Shannon limit.

CONCLUSIONS

Table 1 summarizes the achieved (and planned) MUX factors with different LTD multiplexing techniques, as well as the amplifier bandwidth B, and the estimated Shannon limit on the number of multiplexed channels. The low MUX efficiencies in Table I indicate that existing instruments are far from the fundamental hmit of the number of channels that can be multiplexed. This gap is in contrast to digital communication systems with advanced error correcting coding schemes [32], which can approach the Shannon hmit to within a small fraction of a dB.

In order to fulfil the scientific potential of very large LTD arrays, investment is needed both to increase the bandwidth and to use the available bandwidth more efficiently. The dissemination of GHz readout techniques, and the introduction of CDM and hybrid basis sets have the potential to assist in meeting the goals of future arrays.

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